I'm only looking on my phone, but none of the wood looks intricately detailed or substantial.
If you don't replace with more substantial, wider trim, I'd paint it -- white trim, IMO, always looks crisper, cleaner and fresher. EDIT: I like stained wood trim when it is much wider than what is pictured and matches the floors seamlessly. My home growing up is all stained wood trim, but the baseboards are at least 8" high and the trim around doorways are probably about 6".
I'm simply of the mind that, in the home, there's no reason to keep something a certain way because of principle rather than preference. If you prefer the unpainted wood trim, then by all means figure out a way to make the process of refinishing less painful/time-consuming. But if you prefer the look of white trim, paint that beautiful wood.
The doors I'd spend the time sanding and staining to your liking, again, if that's your preference.
