Based on the style, it's probably oak.....about 1910 to 1925. Better pictures will tell more precisely. Linda C

Solutions will vary by state. Fumigation is the most certain way of killing the bugs, but is nearly impossible to have done in some places (like Massachusetts). Borate-containing products are the ones most widely recommended and available (Bora-Care is one brand name). With a piece of furniture you can inject it into the holes using a syringe and hope that it will permeate the wood to reach the bugs that are left inside (the hole is left when the insect bores its way out and leaves). You can also paint it on any unfinished surfaces. I'd be very leery of bringing infested wood into my house, so would first look into having the piece fumigated.

Marv did something! It looks like you have two problems: the mirror backing and the loose trim pieces. I'm not sure how to deal with either. It depends on how "original" you want to keep it I guess. Small dab of wood glue for the trim bits I guess, after you get it all cleaned up? Don't try to fix all the cracks and make it look perfect. Go for Clean, with Character. To reattach the backing, you'll probably want too pull out the old nails and use new ones in slightly different spots, so with new holes in backing and frame (pilot holes, please!). If the backing is warped, it won't work as well, but you don't want to replace the backing because Marv will haunt you. :) Maybe instead, use regular mirror clip things, as low profile as you can find, and then loosely reattach Marv's backing. You do want that mirror snug inside if you're going to hang it. I don't know about this, though. I'd probably spend some time browsing in the hardware store which we aren't supposed to be doing right now.

Looks like a very nice piece. I would have guessed maple but I'm not a pro. Could you try taking pics without the flash in natural light so that the grain is easier to see? ETA: There's also an Antiques and Collectibles forum - consider cross-posting to that forum.