In small jobs like that, fixing is often more work than tearing out and starting over, especially if the original workmanship is questionable.
If you try to do it a wall at a time, you'll go nuts. Go big and git 'er done, IMO. Plan the electrical, RG-6 (cable/coax), Cat-6 (ethernet/phone) that you'll want and build it right.
I am inclined to agree with you. Especially where there is water intrusion in 2-3 walls of 4 walls. . . Not to mention it appears that he is in the early stages of finding all of the problems, I would anticipate the mushroom factor to appear shortly.
Seeing that the original wood siding is still on the (previous) exterior of the house, I would want to make sure that the room had been correctly connected to the house flashed correctly.
To the original question, yes you get at the plywood from the exterior, the apply your house wrap (or use zip panels and be done with it) flash and reinstall the siding; at least that is what I remember for helping on similar projects when I was much younger.
Before you put a lot of effort into this project, I want to address your original complaint; it gets too hot or too cold. I have an insulated sunroom on this house with double glazed argon filled window facing the exterior on 2 sides and the same style of windows and doors facing the interior of the house, it even has its own heat register (no A/C here). In the winter it is still slightly colder than the rest of the house, but in the summer . . . oh boy we close it off from the house until it is cool enough outside to open the windows for the night and by then it is 15-20 degrees warmer than the rest of the house.
There are 2 items at play for us. First, a lot of windows with a good bit of sun results in a high thermal gains from the sunlight entering (something you want in the cold times, but not the warm ones.) Second, if I recall correctly an energy star window has has an R-value of around 3 and maybe up to 5, older and single pane windows will drop significantly (To put that in perspective lets look at this chart
https://www.archtoolbox.com/materials-systems/thermal-moisture-protection/rvalues.html If you have siding, 1/2" plywood, an air gap greater than 1/2", and 1/2" dry wall on the interior you may already be looking at R-2.68 or so.) So insulating the wall cavities will improve your situation, but if you have a lot of windows you may still be looking at temperatures that are more extreme than you would like.