Our friends Jim and Cybele, the ones we got our stove from, came over and brought a heat gun. The first layer blistered off quickly to reveal multiple layers of paint. (Yes, that is avocado green!)
Jim dug a little further to get to the bare wood. We think it's either pine or Douglas-fir.
Jim thinks our trim is of an intermediary style, a transition between the more ornate Victorian style and the humbler bungalow style. That piece of info tipped us over the edge; we're going to keep it. I bought a respirator today so I can start attacking the trim with the heat gun Jim loaned us.
Don't change that dial -- more bare wood coming soon!