G.O.P. primary victories in Nevada set the stage for Trump-centered battles in the fall.

Republican voters in Nevada on Tuesday elevated conservative candidates who have ardently embraced Donald J. Trump’s false claims of election fraud, turning a key swing state into a contest this fall between embattled Democrats and Republicans who insist President Biden stole the 2020 election.

The victories in the Nevada primaries for Mr. Trump capped a series of elections on Tuesday that saw one South Carolina Republican lawmaker who had crossed Mr. Trump go down in defeat, another survive her Trump-backed challenge and a Hispanic Republican grab a South Texas House seat vacated by a Democrat.

Those results gave mixed signals about Mr. Trump’s continuing grip on the party even as the scrutiny of his actions following his 2020 defeat intensifies. At the same time, the elections on Tuesday suggested that Republicans remain on course for strong gains in November’s midterms.

By flipping the Rio Grande Valley seat of former Representative Filemon Vela in Texas, Mayra Flores became the first Republican to represent the majority-Hispanic district in the seat’s 10-year history, and she became the first Republican Latina the state has ever sent to Congress.

In the sheer number of tossup contests, few states will rival Nevada this fall. Republicans see chances to unseat a host of Democrats, including Gov. Steve Sisolak; Lt. Gov. Lisa Cano Burkhead; three Democratic members of the House; and Senator Catherine Cortez Masto.

Among the Republicans who won their primaries Tuesday were Adam Laxalt, a Senate candidate and former Nevada attorney general who led Mr. Trump’s efforts to overturn the state’s 2020 election results, and State Assemblyman Jim Marchant, a secretary of state candidate who has pressed conspiracy theories about voting machines and hopes to oversee the state’s 2024 election.

Election night on Tuesday started with the defeat in South Carolina of Representative Tom Rice by a Republican primary challenger endorsed by Mr. Trump, even as another South Carolina Republican, Representative Nancy Mace, survived.

Both Mr. Rice and Ms. Mace had crossed the former president as he struggled to maintain power after the Jan. 6 attack, which is now under the spotlight of congressional hearings. Mr. Rice, a staunch conservative in a conservative coastal district, was one of 10 House Republicans who voted to impeach him for inciting the riot. Ms. Mace, in her first speech as a newly elected freshman, said Mr. Trump bore responsibility for the deadly mayhem, though she did not vote to impeach him.

In turn, Mr. Trump backed Katie Arrington to take on Ms. Mace and State Representative Russell Fry to challenge Mr. Rice, who said Mr. Trump has been on a “traveling revenge tour.” Mr. Trump, who turned 76 on Tuesday, called on South Carolina voters to deliver him “a beautiful, beautiful birthday present” — twin defeats of both Ms. Mace and Mr. Rice.

The South Carolina contests had their own dynamics — Mr. Rice was defiant and contemptuous of Mr. Trump to the end, while Ms. Mace tried hard to regain the good graces of Trump administration officials if not Mr. Trump himself. The outcomes of both races could hold deep meaning to the party as it considers whether to renominate the former president for another White House run.

“This took a little bit of time but we are finally here,” Ms. Mace told those gathered for a victory party in Charleston, as she thanked Ms. Arrington for “stepping into the arena.” She added, “this is going to make our campaign even stronger in November.”

The elections on Tuesday represented something of a midpoint in a Republican primary season that has delivered decidedly mixed signals to party leadership. Mr. Trump has claimed some significant wins, propelling his chosen Senate candidates to primary victories, such as J.D. Vance in Ohio and Mehmet Oz in Pennsylvania. However, his endorsed candidates have lost primary showdowns for governor in Georgia and Nebraska as well as a key secretary of state race in Georgia.

Still to come are contests that rank high on his vengeance list, such as Representative Liz Cheney’s primary in Wyoming on Aug. 16. Ahead of Arizona’s Aug. 2 primary, Mr. Trump has backed Kari Lake, a promoter of his false stolen-election claims, to be the state’s next governor. To take on Senator Mark Kelly of Arizona, he picked Blake Masters, who was caught on tape promoting the conspiracy that one-third of the people outside the Capitol on Jan. 6 were F.B.I. agents.

In South Carolina, Mr. Rice was only the second of the 10 impeachment Republicans to take his case for re-election to the party’s primary voters, and he was the first to lose. The other, Representative David Valadao of California, clings to a slim lead over a Trump-aligned challenger, as vote counting continues after the primary there last week. Mr. Rice’s defeat means half of the 10 will not be returning to Congress next year, with other contests still to come, including Ms. Cheney’s uphill climb.

Ms. Mace’s run for re-election had split the Trump community. The Trump administration’s most prominent South Carolinians — former Representative Mick Mulvaney, his budget director and acting chief of staff, and former Gov. Nikki Haley, who served as United Nations ambassador — both backed Ms. Mace against Ms. Arrington, who was a proven gamble for the party since she had lost the seat to a Democrat in 2018.

Ms. Haley, who is considering her own run for president in 2024, had scrambled to endorse Ms. Mace before Mr. Trump could endorse Ms. Arrington, a move that established some independence without openly crossing the former president.

“It’s a great day in South Carolina!” Ms. Haley proclaimed with Ms. Mace’s victory.

In the Republican upset in South Texas, Ms. Flores won a special election to fill the remainder of Mr. Vela’s term until the end of the year, becoming one of three Latinas to ever represent the state in Congress. The seat will once again be up for grabs in the November general election. Yet even her temporary victory foreshadows broader Republican gains in the Democratic stronghold of South Texas.

Ms. Flores — who was born in Tamaulipas, Mexico, and is the wife of a Border Patrol agent — raised 16 times the amount logged by her closest Democratic competitor, Dan Sanchez.

She had not received a formal endorsement from Mr. Trump, but she had campaigned as a Trump-inspired Republican focused on border security. Her campaign signs highlighted three words: “Dios, familia, patria.” God, family, country.

In one of her earliest campaign ads, she made her way through a thick field of flowering cotton in South Texas, as she blasted a Democratic Party that she said insists on selling Hispanics the idea that they should depend on big government.

“At 13 years old, I was working in this very cotton field every day, all day, in the hot Texas sun,” she said, adding that immigrants like her came “the legal way” to pursue the American Dream. She called for a militarization of the border, embraced Mr. Trump’s false claims of election fraud and often denounced the “radical socialist communist agenda.”

In Nevada, Republican candidates largely sought to align themselves with Mr. Trump, taking hard right stances on abortion, guns, immigration and the teaching of race and gender in schools.

In the Senate race, Ms. Cortez Masto will face Mr. Laxalt, the grandson of a former Nevada governor and senator who held off a late surge from Sam Brown, a veteran.

But it was Mr. Marchant who worries many Democrats the most. At events with the MyPillow chief executive Mike Lindell and other Trump allies, Mr. Marchant has embraced some of the most far-fetched and debunked electoral conspiracy theories. He has pressed for all ballots to be cast and counted by hand, and he organized the “America First” secretary of state coalition to elect candidates who have embraced false claims about the 2020 election.

As it has been in the past several elections, Nevada promises to be a battleground in the 2024 presidential campaign, and the person running the election will either be Mr. Marchant or Cisco Aguilar, a Democrat and former aide to the late Senate majority leader Harry Reid.

“Enough of the untruths. We need to focus on the truths in a bipartisan way,” Mr. Aguilar told The Las Vegas Review-Journal.

Beyond those statewide offices, three Nevada House seats are deemed tossups for the fall, a potential windfall for Republicans as they seek control of Congress.

In Nevada, where Mr. Reid and the state’s Culinary Union built an influential Democratic political machine, a racially diverse coalition of working-class and Latino voters has powered crucial Democratic victories in presidential elections since 2008. But the president’s party tends to lose ground in midterm elections. That has been particularly true for Democrats in Nevada. Democratic turnout in the midterms in the state has tended to take a sharp nosedive, favoring Republicans.

Maya King and Jennifer Medina contributed reporting.