Making his debut onstage, he was greeted by forceful challenges from Elizabeth Warren and the rest of the Democratic field. Michael R. Bloomberg bobbed behind his lectern, as if the motion might deliver him somewhere more comfortable. He blinked, then blinked some more. He appeared unsteady — for all the preparation his billions might buy him — on questions of race and gender that could not have come as a surprise. Pressed about allegations of a hostile workplace at his company, Mr. Bloomberg wandered into a legalistic defense of nondisclosure agreements, adding that perhaps women “didn’t like a joke I told.” Questioned on his longstanding support for stop-and-frisk policing, a signature policy of his mayoralty in New York, he professed himself “embarrassed” before suggesting others onstage also had plenty to apologize for. “Remember,” he said in one exchange, explaining why he had not yet released full tax documents. “I only entered into this race 10 weeks ago.” That much was clear. Until Wednesday, as Mr. Bloomberg spent heavily and campaigned atypically, bypassing the early-voting states to focus on delegate-rich contests in March and beyond, his candidacy had existed almost in parentheses: Former Vice President Joseph R. Biden Jr. was the early front-runner. (But Mr. Bloomberg wanted his voters.) Senator Bernie Sanders won New Hampshire. (But would Mr. Bloomberg’s fortune bury him by the spring?)