I wonder if he would have publicly supported Adams had he lived, so that he would have been re-elected in 1800. But then again, Hamilton was against Adams at this point, so Washington might have been also.
Washington would not have picked sides one way or another. He'd have stayed out of it.
This is incorrect. For all his talk about nonpartisanship, Washington had become quite the Federalist partisan by the end of his life. He spent his last couple years trying to use his influence for the benefit of various Federalist candidates, and he and Jefferson were not on speaking terms at the end of his life.