Does the Chief Justice not preside? Chief Justice William Rehnquist presided over Clinton. But I've heard (can't remember the reason) that perhaps Roberts would have to recuse.
#1252 b/c Roberts oversees the FISA courts
I took the question as a general one, as in who picks the judge in “any trial.” Your question is addressed to one particular type of trial, that being an impeachment trial of the president in the senate.
The constitution directs that impeachment trial of the president will be presided by CJ of SCOTUS. There is no provision or wiggle room for recusal, but judges don’t follow the law or constitution anyway, so who knows.