There's a lot in those questions. The role of the President is to defend the Constitution and faithfully execute the laws written by Congress. He has the ultimate power to order an investigation take place, and to order one to be stopped. He could order the DOJ and/or the FBI to do a thorough investigation of Hillary Clinton, for example. He could order the investigation into Flynn, or the Russian meddling in the election be stopped cold. Those receiving the orders can either comply or resign.
The are political ramifications for personally directing investigations, including everything from losing an election to being impeached for abuse of power (which would certainly happen if a president ordered an investigation of all his (or her) political opponents). The president could, for example, order the FBI (or the IRS) to get all up in Jake Tapper's business. Many presidents have pulled those kinds of stunts, and the eventual political fallout is usually not good. But it's within the president's constitutional authority to do so.
Trump could call up the FBI and say, "The Flynn investigation is over. I've just given Flynn a pardon." And the investigation ends right there. The political backlash would be epic.
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