Not a pro but I think it'd be a lot of effort for very little reward.

I like the wood trim. I say leave it.

I think painting the trim and baseboards white would be very cool !

Keep the wood trim around the doors. You could paint the baseboards white or keep them as is. If you decide you want the baseboards white it would be worth it to have them replaced with prepainted larger ones rather than having to sand, prime, and paint the ones you have.

Leave the wood trim as is to match the lovely doors. New lighting will make a tremendous difference.

Yes I definitely want to get big thick white baseboards everywhere but Im trying to not spend money right now and just do something to make it a little better. I already have a 5 gallon bucket of white paint and could do this project with little expense and in a couple years do all new.