While waiting until fall will let you see if there would be more recovery, and you can wait until the tree goes into dormancy, I don't think you need to wait that long. I would wait a few more weeks, though, so there will be no sap running, as the tree is pretty stressed and doesn't need to have a cut weeping sap out onto the soil. Clean out the obvious dead wood, being careful not to damage the new growth. A sharp, by-pass pruner will help, as will a narrow pruning saw for larger branches. If you are in doubt about the viabilty of a branch, you can lightly scratch the bark, and see if the cambium layer just under the bark is still alive - it will be green if it is. If it's grey or brown all the way to wood, then the branch is dead. If it snaps easily, it's dead. If it just bends, it may still be alive. Cutting, however, is safer than breaking off the dead wood - it won't necessarily break off where you want, and may damage new wood lower down. Don't try to clean it all out in one day - you can always come back on another day and cut out more, but you can't put back what you have cut off. Start with the obvious things, and go slowly. It's very easy to slide over and cut HERE when you meant to cut THERE. If you can post a photo or two (three?) of the seemingly dead stuff, then you can get advice on where to cut if there is any question. It's usually pretty obvious. The Maple Forum is mostly about Japanese maples, so you can also post there and ask for advice.

That plant is ALIVE!! I have two foot cuttings that I put in the ground a year ago that are so dead above ground that you can snap pieces off with your fingers but as soon as the temperature got in the 70's last week, sprouts are growing out of the ground at their base. Fig tree cuttings are incredibly tenacious to life. Once you put a cutting in the ground, don't go cutting all the dead wood off it months later just to see if it is alive. Just leave it alone and wait for hot weather to wake it up. You should not have cut the dead wood off that tree this early. Be patient in the future and do all wood cutting in the late fall. I have yanked out what I thought to be completely dead cuttings only to find a clump of roots at the base. I would coat the area that you cut with some kind of sealer. Have faith in your fig tree. It will grow.

Those 2 towering branches can be taken out which should force more energy into the lower branches, encouraging them to get taller. If you do that, I would suggest grafting them to herbaceous roots just so you don't end up wasting very good scion wood. Even branches with no buds can be forced into producing a bud to create a new plant. Grafting is easier than most people would think it is, and should be done best in the fall. If you cannot get a hold of any herbaceous peony roots for the grafting, then I would suggest simply taking those branches and forcing them to root. I've successfully rooted some Japanese Tree peony cuttings. I admit that they aren't that vigorous, but it works if you give them the right conditions. I think using a broad fork to aerate the soil a bit, adding a few lbs of gypsum & azomite, and a layer of compost should rejuvenate this peony for next year's growth. I believe this is the species you have right here as the flowers and foliage look identical to yours. This is a Yagumo japanese tree peony. Those top leaves are definitely staying on without any die back so conditioning the soil should help yours regain its health.

Any stored clothing, as in woolens? Could be carpet beetles. But that's just a guess until you can post close-ups. Adult insects go toward the light because they "know" it's the way out. The problem is that no one told them about windows.