They will be fine. Sand them very well, and fill in the nail holes. If restaining, you may want to use a bit of stripper on them too. If painting, then prime and paint 2 coats of high quality trim paint. I am hoping you go through the extra effort of staining. I also wouldnt replace the pieces of wood, unless, they were too bad to save, because, the new wood, is not anywhere near the quality of what you already have, and it most likely wouldnt stain up to match the older wood. Remember to work in well ventilated place, and wear a mask, and having things checked for lead or asbestos before you start sanding and tearing out anything. Congratulations on your new home.

Are you sure that's drywall and not plaster? If it's drywall then that explains it being recessed as there was no such thing as drywall back in the early 1900's.

You'll never be able to replace the wood because there is no longer old growth wood like in your house.

I'd clean it well and coat it with a product like MW Polyshades. There are plenty of suppliers of old growth wood, many with prices comparable to new growth wood.

It looks like they put sheetrock on top of plaster this why the trim is flush. If you start taking this apart it will be a big mess.

Being this flat trim, if you want to leave existing molding, you can use a backband molding along the outer edges to clean up the corners but you will need backband molding with a low profile since existing molding is recessed into a wall covering...You can get a peace of molding just to see how it fits. This is a few common sizes of backband moldings

Another way is to nail same width trim over it and use backband molding, this way you will have new and clean moldings throughout and you can finish it as you like and it will look like this, this is a plain backband molding, but you can get it with a more decorative profile like you see in the picture above.

Good luck

Please dont use poly shades, that stuff is just not very good stuff. It will cover the grain of your old growth wood. Please sand, them, or strip them, and sand them, and put on regular stain, with regular poly for a top coat.