SPARKS, Nev. — Joseph R. Biden Jr. called an octogenarian voter a “damn liar” and challenged him to a push-up contest. He dismissed a heckler as an “idiot.” He commanded the news media to focus on President Trump instead of the overseas business dealings of his son, Hunter Biden, demanding of one reporter, “Ask the right question!”

For months now, Mr. Biden has been confronted on the campaign trail with questions, attacks and misinformation concerning his son — encounters that have taken on a dramatic feel, given the uncertainty of how Mr. Biden will respond. As he began to address the crowd in a high school gym in Sparks this month, a group of protesters held up letters spelling out a taunt that Mr. Trump uses regularly: “Where’s Hunter?” Mr. Biden responded by saying his son “sends his best regards.”

As the Senate impeachment trial of the president continues this week, there is renewed focus in Washington on Hunter Biden, who held a seat on the board of a Ukrainian energy company at a time when his father was vice president and handling diplomacy with the country.

And Mr. Biden is no longer just dealing with questions about his son from hecklers: On Wednesday, he rejected the suggestion that he and his son testify in the trial in a swap for the testimony of current or former Trump administration officials, an idea raised by an attendee at an Iowa campaign event that has been dismissed by congressional Democrats. “This is a constitutional issue, and we’re not going to turn it into a farce, into some kind of political theater,” Mr. Biden said.