Is it typical for water to stay standing in the P trap? I would guess
so, lots of rookie questions I have here.


Yes, it's normal for water to stay in the P trap. That's what it's
for. If this is an ongoing problem, one half-assed solution is to
keep a gallon of RV antifreeze nearby, and pour about 8 oz down the
drain on cold nights, after you're done with everything else.