Pp here--a good way to test if the wood needs conditioning is to press your fingernail into it. If it makes a mark, it's soft wood like pine and it helps to use wood conditioner before staining. If its an older frAme, it's probably hard wood. New stuff is more likely to be cheap quick grow soft woods (which is why the new cribs break after a few years, but that's a different post...)

You need to take the picture out and strip the current finish, either by sanding or with a chemical. Use very good ventilation. Then stain, then seal with polyurethane or an oil like tung oil. Don't use the combo stain and sealant--it stinks.
Painting it would be somewhat easier--just rough up the current finish with sandpaper so that the paint will stick. Use a good brush or you'll have brush marks.

It depends on the wood species, which I'm guessing you don't know. Some species are prone to blotching and require pre-sealing. Wood staining can be more complicated than people think.

Can you remove the artwork from the frame, then refinish the wood frame and reinstall the artwork in the frame?

Are you talking about painting the frame, while the art is inside? I would worry about wrecking the art.

Yes as long as it doesn't have a shiney coating. Can easily do annie sloan chalk paint but those are colors not stains but it'll go right on easily.

I have a painting that is currently custom-framed in a wood finish that I don't like. It's an odd sized painting, so finding a new frame seems tricky, and I'd rather not pay the expense of having a new one custom made. I am the least crafty person on this planet...so my question might sound really dumb, but I truly don't know these things. Can I buy a wood stain and just paint the frame?

Right now the frame is a light, ashy color. I would like it to look more dark brown/mahogany.

Thanks for any advice.