When Ms. Harris took the stage, he lifted up her arm like a prizefighter in celebration. She quickly seemed to try to adopt his mantle, leading the audience, the largest she has drawn since becoming the Democratic nominee, in a chant of “Yes, we can,” Mr. Obama’s 2008 campaign slogan.

“Millions of Americans were energized and inspired not only by Barack Obama’s message but by how he leads,” Ms. Harris said after he ceded the lectern to her. “Seeking to unite rather than separate us.”

She proceeded to attack former President Donald J. Trump as an “unserious” yet dangerous authoritarian who would hurt Americans in their everyday lives even as he undermined the nation’s democracy.

This year, the Harris campaign is hoping to use the power of the party’s most popular politicians and major celebrities to energize its base voters. Mr. Obama and Ms. Harris were joined on Thursday by the rocker Bruce Springsteen, who played a three-song set with guitar and harmonica before she spoke, accusing Mr. Trump of running to be an “American tyrant.” On Friday night in Houston, Ms. Harris is set to appear with Beyoncé, one of the world’s most popular musicians. On Saturday, she will rally with Michelle Obama in Michigan.

The breadth of Ms. Harris’s celebrity entertainers on Thursday spoke to the coalition she is trying to build. There was Mr. Springsteen, who is especially beloved by older, white Americans; the actor and comedian Tyler Perry, who is popular with Black women and gave an impassioned speech about growing up in poverty; and prominent Black entertainers including Spike Lee and Samuel L. Jackson, who proudly noted to the crowd that he and Ms. Harris share a favorite curse word.

Jake Schneider, the Trump campaign’s rapid response director, dismissed their impact on voters, saying that “relying on celebrities is nothing new for the party of Hollywood elites.”

Ms. Harris and Mr. Obama are longtime friends, and she endorsed him over Hillary Clinton in 2007 when she served as the district attorney of San Francisco, bucking most of the Democratic Party establishment.

Georgia is a top battleground state and Atlanta, with a significant Black population, is its biggest driver of Democratic votes. But the vice president’s support among Black voters, especially Black men, has been lower than is typical for a Democrat running for president. This month, Mr. Obama suggested sexism was to blame. On Thursday, one of Ms. Harris’s introducers, Senator Raphael Warnock of Georgia, tried to pour cold water on fears that Black men would support her opponent.

“We’re not confused,” he said. “We know that this is the man who was held accountable by the Justice Department because he wouldn’t even rent apartments to Black people,” Mr. Warnock said.

In 2020, Joseph R. Biden Jr. won Georgia by fewer than 13,000 votes, the first time a Democrat had carried the state in a presidential election since 1992. His victory — powered, in part, by demographic shifts and a concerted Democratic effort to reach new voters in the state — has left the party dreaming of a repeat this fall. But polls show an exceedingly tight race.

Both campaigns have responded by pouring resources into Georgia, where more than two million people have already voted. Mr. Trump visited on Wednesday for the second time in eight days, and Ms. Harris’s trip on Thursday was her second in a week.

As it has elsewhere, Mr. Trump’s campaign has focused on the economy and immigration in Georgia. As he tries to provoke fear about the increase in migrants crossing the border during much of the Biden administration, the former president has frequently cited the death of Laken Riley, a 22-year-old nursing student who the authorities said was killed by an undocumented immigrant.

The Harris campaign has emphasized abortion rights as a top issue in Georgia, which bans the procedure in most cases at about six weeks. On the trail, Ms. Harris and her allies have told the stories of Amber Thurman and Candi Miller, two Georgia women who died after treatment delays that stemmed from the ban, according to reporting by ProPublica. Now Ms. Harris is trying to win the support of moderate independent and Republican voters by blaming Mr. Trump for the abortion bans passed in many conservative states.

“In every state in the South, including Georgia, there is a Trump abortion ban,” she said on Thursday. “Many with no exceptions even for rape or incest.”

Although Ms. Harris was well received in Clarkston, there were dangers in following Mr. Obama, one of the nation’s most gifted political orators.

As she spoke, some members of the crowd, who had waited for hours in the heat, started trickling toward the exits.

Reporting was contributed by Michael Gold , Alan Blinder , Maya King and Rebecca Davis O’Brien in Clarkston, Ga., and Erica L. Green in Washington.

Yet Ms. Adelson, a physician and a conservative megadonor, and her operatives have been eager to keep Mr. Trump on television in the battleground states of Wisconsin and Michigan, especially given that he is being outspent by Vice President Kamala Harris and her allies. So other donors have said that beginning in early October, Ms. Adelson has been soliciting other billionaires to help bridge the gap to keep the group on the air through Election Day.

“We had an initial $100 million,” said Dave Carney, a senior adviser to the group. “We’re trying to raise more, and Dr. Adelson has been a fund-raising star getting more people on board.”

Super PACs formed by a single rich donor can struggle to raise outside money as fellow billionaires wonder why the patron won’t foot the whole bill. Ms. Adelson has raised over $10 million for her super PAC over the last few weeks, a spokesman for her said. Supporters of the group who will be made public in a Thursday filing with the Federal Election Commission include the conservative billionaires Liz Uihlein, Ronnie Cameron and Diane Hendricks, who gave $3 million, $2 million and $1 million, respectively. Mr. Carney said the group now had over 100 donors.

Ms. Adelson has also been collaborating on raising money for another pro-Trump super PAC, MAGA Inc. On Thursday, one person who considers Ms. Adelson to be a mentor — Jan Koum, the billionaire co-founder of WhatsApp — disclosed having given a $5 million check to MAGA Inc. earlier this month.

Ms. Adelson was planning to host an event for MAGA Inc. in Las Vegas earlier this month that would have featured Dana White of the Ultimate Fighting Championship and Mr. Trump, but the groups called off the event because it came too close to the Jewish holiday of Yom Kippur. Some donors had been asked for $1 million contributions in connection with that event.

Ms. Adelson has emerged as one of Mr. Trump’s strongest supporters, even though Mr. Trump has not always seemed especially appreciative. Mr. Trump told associates earlier this year that he expected Ms. Adelson to donate an extraordinary $250 million to back him — an expectation that it appears she will not meet.

The ads, titled “A Warning,” are scheduled to immediately go into the Harris campaign’s rotation of television and digital advertising, a campaign official said, adding that they would be targeted in particular at markets with larger populations of veterans.

And Ms. Harris underscored the message at a CNN town hall on Wednesday when she herself also called Mr. Trump a “fascist.”

Both the 30-second and the 60-second ads begin with a black screen and a pulsating, alarm-like sound as the words, “An unprecedented warning …” are typed onto the screen.

The text then identifies Mr. Kelly, Mr. Trump’s former chief of staff and a four-star Marine general, before cutting to a recording of Mr. Kelly’s recent interview with a reporter for The New York Times, Michael S. Schmidt.

“Do you think he’s a fascist?” Mr. Schmidt asks.

The 30-second version compresses Mr. Kelly’s response: “He certainly falls into the general definition of a fascist: using the military to go after American citizens.”

The 60-second version quotes Mr. Kelly at greater length: “He certainly falls into the general definition of a fascist. It’s a far-right, authoritarian, ultranationalist political ideology and movement characterized by a dictatorial leader. The former president — he is certainly an authoritarian. Using the military to go after American citizens is a very bad thing.”

Both ads also include a clip of Mr. Kelly quoting Mr. Trump as saying, “Hitler did some good things, too.”

The pulsating alarm sound continues throughout both ads, with the same red text typing at the end: “Donald Trump is unhinged. Unstable. In pursuit of unchecked power.”

Mr. Trump has attacked Mr. Kelly since his public comments, such as in an interview on Fox News on Thursday in Arizona.

“I fired him,” Mr. Trump said, according to a transcript provided by Fox News. “He made a statement that I’m like Hitler. It’s — just couldn’t be further from the truth. It’s just the opposite, actually.”

It was unclear whether the man, Dieter Klofkorn, 35, had a lawyer. The Phoenix police said he told investigators that he had set the postal box on fire because he wanted to be arrested. He was charged with one count of arson.

The U.S. Postal Inspection Service said it had taken possession of the damaged ballots. Michael Martel, a spokesman for the service, said in a statement that inspectors were working to “ensure any affected election mail is remedied.”

Arizona’s secretary of state, Adrian Fontes, said that the voters whose ballots were damaged would be contacted to make sure that they would be able to get a new ballot.

The Maricopa County Recorder’s Office, which is responsible for mail-in voting and voter registration in Arizona’s most populous county, which includes Phoenix, encouraged voters to check the status of their ballots through an online portal. Friday is the last day that Arizona voters can request replacement ballots, according to the office.

Arizona, which could be decisive in this year’s presidential election, has been a hotbed for election conspiracy theories since Joseph R. Biden Jr. defeated Donald J. Trump there in 2020, flipping the state, including Maricopa County.

Arizona has also been vexed by threats of election-related violence.

In Tempe, near Phoenix, a Democratic Party campaign office was closed after being damaged in three shootings across four weeks. (No one was hurt in the shootings.) A 60-year-old Arizona man, Jeffrey Michael Kelly, was arrested on Tuesday in connection with the shootings and charged with three counts of terrorism.

Neha Bhatia, a lawyer with the Maricopa County Attorney’s Office, said in a court hearing on Wednesday that Mr. Kelly had more than 120 guns, 250,000 rounds of ammunition and a grenade launcher at his home and that the authorities believed he had been “preparing to commit an act of mass casualty.”

This week has seen a small but growing concern about an extreme weather event around Election Day, and that fear has been amplified by people with large followings on social media. But the storm’s shape, timing and very existence are still very much hypothetical.

The illustrations show just one of several possible outcomes. They’re also predicting an outcome nearly two weeks away, which in the meteorological community is considered “guidance.” While the computer models used in forecasting weather have improved greatly over the years, generally anything over a week away isn’t something you should change your plans for — yet.

An actual forecast is more than a single image of a computer weather model. It’s an expert hypothesis based on all the available data, and meteorologists will look at trends in the data when they draw up a forecast. The global models used by experts, the Global Forecast Model and the European model, are updated four times a day, and each time forecasters will consider what has changed and what has been consistent from one to another. From this, and with the passage of time, they start to draw a likely forecast.

On Thursday afternoon, the American model runs continued to show a storm forming in the Caribbean, moving into the Gulf of Mexico and approaching Florida on Election Day.

Other forecast models, though, showed nothing like this potential storm. A concern troubling forecasters on Thursday was that the other models did show the type of weather conditions that could be favorable for some tropical activity in the Caribbean Sea and the Gulf of Mexico during the same time period.

This also isn’t surprising. If a hurricane were to form at this point in the season, the Caribbean Sea is where meteorologists would expect it to happen.

The Climate Prediction Center issues probability forecasts two weeks out, and it, too, has recently shown an increased likelihood of tropical activity in the Caribbean — exactly where the forecast model does show a potential storm.

So the likelihood of something forming in the Caribbean Sea toward the end of October and ahead of Election Day is high, the center’s forecasters said. But whether that “something” is a rainstorm, a tropical storm or a hurricane — and the path and timing of any system — will become clear only in the coming week.

Mr. Brown, who also ran for the Senate and lost in his party’s primary in 2022 when Senator Catherine Cortez Masto was up for re-election, has struggled to gain traction against Ms. Rosen, a former U.S. representative finishing her first term in the Senate. She has pummeled him with television advertising highlighting his past opposition to abortion and other positions.

The Senate Leadership Fund has focused its financial firepower on Senate races in Montana and Ohio, where Republicans see prime pickup opportunities. Democrats hold a 51-seat Senate majority, and winning in Nevada could help solidify Republican control of the chamber.

Public polling has shown Ms. Rosen with a consistent lead, but the presidential contest in Nevada, a critical battleground state, is essentially tied. Republicans hope that former President Donald J. Trump’s supporters and voters’ frustrations over the high costs of basic goods can lift Mr. Brown to victory.

“Jacky Rosen has been a reliable vote for the Democrats’ extreme agenda and is a lackluster candidate,” Steven Law, the president of the Senate Leadership Fund, said in a statement. “President Trump is doing very well in Nevada, and we think Sam Brown can, too.”

In a statement, Johanna Warshaw, a spokeswoman for the Rosen campaign, called Ms. Rosen “one of the most bipartisan, independent and effective senators” and said that Mr. Brown was “a failed politician whose struggling campaign has been losing support from all sides thanks to his tenuous ties to Nevada, his extreme MAGA positions, and his disqualifying anti-abortion record.”

Since July, Democrats have spent close to $65 million supporting Ms. Rosen. Republicans beyond the Senate Leadership Fund have poured about $40 million into the race, according to AdImpact, a media-tracking firm. Last week, the candidates squared off in their only televised debate, which proved heavy on policy. Both candidates appeared relatively mild-mannered.

Throughout the race, Mr. Brown has closely aligned himself with Mr. Trump, showing up whenever the former president has campaigned in Nevada. His hopes ride on Mr. Trump performing well in the state.

“We’re a team in that we both could have a totally different life,” Mr. Brown told The New York Times before a recent rally for Mr. Trump in Reno, Nev.

“I don’t need this,” he added. “President Trump doesn’t need us. People right now need that sort of selfless leader.”

Hours after the hearing concluded, Ms. Sykes responded with defiance, casting her vote early in the very district at the center of the dispute.

Given the board’s inability to act and that Ms. Sykes’s ballot has been cast, Ohio’s Republican secretary of state will be tasked with making a final decision.

A Republican political activist initiated the challenge to Ms. Sykes’s residency. It came in the final days of a tight re-election race in which she faces Kevin Coughlin, a former state lawmaker. He has been quick to try to capitalize on the case in an apparent effort to damage Ms. Sykes’s standing with voters in the district.

“If she can’t even vote for herself, why should she expect you to?” Mr. Coughlin said in a social media post.

Ms. Sykes dismissed the attacks.

“I am a proud daughter of Akron and I am grateful to call this community my home,” she said in a statement. “Anyone who says otherwise is purposely spreading a deeply offensive lie for political purposes.”

The Constitution does not require that a member of the House reside in the district they represent — only in the state — and some members of Congress do not. Ohio law also has no such rule, but the case has given Republicans a new line of attack in the final days of the campaign.

Ms. Sykes did not attend the hearing on Thursday but filed a detailed affidavit aiming to answer questions about her longstanding status as a resident in Akron. Along with her statement, she shared copies of her driver’s license, congressional pay stubs and housing documents showing her address in the city.

The issue traces back to an ethics disclosure form submitted earlier this year by her husband, Kevin Boyce, a Columbus-based county commissioner, which erroneously listed her as living with him in Columbus. Mr. Boyce later corrected the mistake, telling election officials the couple maintains separate residences.

“Her husband, who as we all know, is also a public official and a lifetime resident of Columbus, maintains a residence there. She maintains a residence here,” Ms. Sykes’s attorney, Don McTigue, said during the hearing. “There’s nothing wrong with that — for spouses to have different residences.”

The procedural timeline makes it unlikely that the question will be resolved before the Nov. 5 election. Both Republican and Democratic members of the election board have up to two weeks to send letters to Ohio Secretary of State Frank LaRose, all but guaranteeing that the controversy will persist well beyond Election Day.

“When Dad gets home, you know what he says? ‘You’ve been a bad girl, you’ve been a bad little girl, and you’re getting a vigorous spanking right now,’” Mr. Carlson said. Grinning, he went on: “And no, it’s not going to hurt me more than it hurts you. No, it’s not. I’m not going to lie. This is going to hurt you a lot more than it hurts me. And you earned this. You’re getting a vigorous spanking because you’ve been a bad girl.”

The crowd went wild.

Mr. Carlson’s speech — at a rally hosted by the conservative group Turning Point Action that featured Mr. Trump as the headline speaker — was full of disparaging comments about women. He called former Representative Liz Cheney “Dick Cheney’s creepy little daughter” and described Democrats as “the party of weak men and unhappy women, one of which leads to the other, by the way.”

He raged against Vice President Kamala Harris in particular, referring to her as “Carmela or whatever her name is” and asking: “Kamala Harris, who couldn’t change the tire on your truck, much less drive it — how did she wind up at the top of the pyramid? And then once she’s there, she lectures you like you did something. It’s too much! We can’t allow that!” Making a grabbing motion, he described her as a pawn of the Democratic “machine” and called her “just a hapless victim who happened to be there in the right color, so they grabbed her.”

Mr. Carlson told the crowd that they had been unfairly demeaned by Democrats who called their views fringe, exhorting them to view themselves instead as a trampled majority, “the most mistreated group in this nation.”

He cast Democrats as illegitimate, calling them “the most parasitic, useless, violent, nasty, aggressive people in your country.” He said that “they have no legitimacy in a democracy, where the government must rule by the consent of the governed.”

In an apparent reference to people who fought for the Confederacy during the Civil War, he continued: “They tore down statues to their memory. People who never built anything in their lives, they went out of their way to humiliate you and spit on you and the graves of your ancestors.”

And he told the crowd directly that they should not accept the election results if Ms. Harris wins.

“At the end of all of that, when they tell you they’ve won, no!” he said. “You can look them straight in the face and say: ‘I’m sorry, Dad’s home, and he’s pissed.’”