Simple answer as you asked for but you asked two questions at the same time.
So, no and yes.
At least let the jars get to room temperature at the very least.
They are already acidic enough the important pert is that they seal.
It is puree it isn't like you are going to hurt it by processing longer.
Folks years ago didn't have refrigerators what do you think they did with jams and jellies when they opened them up for use.
Worth