Here is the painted sections

Currently two piece base wanting to restore to original three piece and then refinish all to match.

You have beautiful and rather unusual molding with the curved sections in the doorways. I'd first determine what species of wood you're dealing with. Oak was popular in the era of your house, but so were some other species. Oak has the advantage of being resistant to splitting if you choose removal. The disadvantage of oak (and mahogony) is that it is an open grained wood and unless treated beforehand allows paint to penetrate deeply into its pores. This makes stripping a real chore. You might want to try removal in an inconspicuous to see how it goes. If it's very difficult or the wood is prone to splitting, stripping in place is really the only way to go. It would be a real shame to seriously damage this molding. It will be a major project best done a bit at a time. Best of luck.

Looking at the original surviving varnished part, it does not have the characteristic grain that oak would exhibit even through such a dark varnish finish. Based on the date it's really likely it could be red gum, but it could as well be pine.
If the other rooms that are now painted have been so forever, it's probably not going to be worth the effort to wind up with a less than original finish; if paint goes on bare wood, it's never going to match the old stained work.
Grab a heat gun and do some test patches; if the paint quickly comes off revealing a dark varnish, you're all set.
I prefer to take down small moldings for stripping. If you're careful, it's so much more efficient working them on the bench and not so much on the walls, low or high. But you will disturb wallcoverings and perhaps plaster. It's definitely the better job with less painful work (stripper mess running down your elbows).
Casey

Do NOT try to take down those door frames or any othr molding with curved sections to work on them. You will never get them back onto the wall as nicely as they are. You can see the 1/4 circle chunk in the first picture - these were usually installed then touched up with some sort of a molding-making tool afterwards to get perfect alignment.

As for "original finish", you may be looking at it in the bedroom. It was common for houses to have a mix of painted and stained/varnished wood trim ... the decorating columns in the magazines of the era often show painted trim in the halls and bathrooms, with stained in the public areas.

Be aware that "paint grade" moldings and high-coverage stains to mimic the expensive mahogany were widely used then, so under that paint you may have some less than top-grade wood. It will be knot-free but may have no grain to speak of.

Thanks for the tips I'll see about the heat gun and try to remove done paint. Yes, in regards to the wood type I'm at a loss as well, on the back side of the nook door it seems the wood has not been finished and appears to have a red hue. I'll see about getting a few photos to help determine, the house is in Northern California could the wood be redwood?

In Northern California, it definitely could have been redwood. It was local and inexpensive, compared to the imports like oak. Alder was also popular because it's easy to work and stains easily.

Can you post a closeup of the wood for color and grain?

Here are a few photos

Another

Door and door jamb are fir, finer-grained smaller pieces idk; pine, cypress, redwood? For me the most reliable test is to make sawdust, as they all have _very_ distinctive odors, as do many species of wood.
Casey

As far as determining the finish here is a process I got from one of Jeff Jewitt's posts that I've used

Find an inconspicuous spot and dab a little denatured ethyl alcohol on the finish with a small brush or cotton swab. After thirty seconds tap the area with your finger - if it's sticky the finish is shellac. If the alcohol doesn't affect the finish try a little lacquer thinner. If the finish still isn't sticky, it is probably an oil-based varnish or polyurethane.

As Casey said, looks like fir and ??? for the finer grain. It was stained or shellacked to look like the more expensive mahogany, so when you strip it, be ready to deal with the whims of staining fir.

Given the location, alder is likely for the other wood because it was close to local.

Thanks for the help, I'll remove a piece of the base and give it a cut to determine the wood and include a few more close up photos.