The open-ended queries appear to be an attempt to penetrate the president’s thinking, to get at the motivation behind some of his most combative Twitter posts and to examine his relationships with his family and his closest advisers. They deal chiefly with the president’s high-profile firings of the F.B.I. director and his first national security adviser, his treatment of Attorney General Jeff Sessions and a 2016 Trump Tower meeting between campaign officials and Russians offering dirt on Hillary Clinton.

But they also touch on the president’s businesses; any discussions with his longtime personal lawyer, Michael D. Cohen, about a Moscow real estate deal; whether the president knew of any attempt by Mr. Trump’s son-in-law, Jared Kushner, to set up a back channel to Russia during the transition; any contacts he had with Roger J. Stone Jr., a longtime adviser who claimed to have inside information about Democratic email hackings; and what happened during Mr. Trump’s 2013 trip to Moscow for the Miss Universe pageant.