Does this also include no affairs when he was Governor of Arkansas too? Because that could mean he would have ran in 1988.
But just getting rid of the Flowers and Lewinsky scandals and Paula Jones lawsuit alone could have dramatically altered his entire presidency. But primarily his second term. Probably would have given him more time to focus on the Middle East and the al-Queda threat.
Gore would have most certainly picked another running mate and won because as others have said, he would have ran on the Clinton economy more and the peace and prosperity of the late 90s.
George W. Bush remains Governor of Texas until late 2002 and then returns to the private sector. Maybe becomes Energy Secretary for a future Republican president.
Hillary Clinton would have still ran for public office (probably senate) but would have waited until 2002 or 2004. Maybe runs for Governor against George Pataki in 2002 or 2006. Gore's record as president will determine her electoral success.
The thing about Hillary's senate run is 2000 was that was really her only shot for running for and easily winning an open seat for high office in New York. Chuck Schumar already held the other senate seat and unseating Pataki for governor would have been difficult for her. Worst case scenario for Hillary a President Gore nominates her to a cabinet position like Secretary of Education or HHS, nominates her to be a federal court judge (eventually making her a candidate for the Supreme Court), or gives her an ambassadorship (probably Japan or to the U.N.).