From its debut on Oct. 7, 1996, the network, under the tutelage of Roger Ailes, did its share of straightforward reporting but also unmistakably filtered major news stories through a conservative lens. Evening programming, which embodied the Fox News brand, was dominated by right-wing commentators like Bill O’Reilly and Sean Hannity, who hurled opinions and vented resentments with a pugnacity that reflected their boss’s own combativeness. For loyal viewers, it was the network of choice to hear repeatedly about the moral failings of Bill and Hillary Clinton, questions about Barack Obama’s birthplace, doubts about the patriotism of American Muslims, grumblings about the war ostensibly being waged on Christmas, and warnings about “death panels” that would supposedly flourish under the Affordable Care Act, known as Obamacare. After the terrorist attacks of Sept. 11, 2001, Fox News embraced the American flag as if it were its own, then became an uninhibited booster of the Iraq War and an unabashed critic of those who opposed it. The network also made itself a haven for Republicans who had fallen from political grace and hoped to restart their careers, among them Newt Gingrich, Mike Huckabee, John Kasich, Sarah Palin and Rick Santorum. Mr. Hannity’s show was effectively a public-relations vehicle for one presidential candidate who made it to the top: Donald J. Trump.