Well the demo is finished. It turned out to be a mix of old mid-1940s slats covered with plywood for about 8 inches on one side. The rest was plywood covered with ruined fiberboard. The slats looked to be fine and ended on a joist so I think I will leave them. They will be covered with another layer of high quality 3/4" anyway. The floor joists themselves turned out to be 2X10s and looked to be in wonderful shape even around the lavatory!
I ran into 2 problem areas with the flooring though. One side of the room has a fiberglass tub/shower combo. Both layers of (good) plywood run underneath the tub but there is some dry rot on the outer end where it meets the joist. I went ahead and cut everything back to 2 inches from the tub on the bottom layer of ply (sheathing?) and about 1 inch out on the top (subfloor). There are beams running perpendicular to the tub every 13 inches but I think I should hang some 2X6" joists parallel so the ends of that ply rest supported.
Also there is a water heater in the room enclosed in 1/2 plywood walls. The floor inside is in decent condition as there was no vinyl covering to hold in moisture. There is some overhang on the bottom side of it as well as a lack of support on the right that has me concerned so I am thinking of adding some joists here as well.
Here is a quick little diagram to show the condition of things:
Will this provide enough support for everything?
Also for the final floor prep, I am going to fill any voids/irregularities with "Well-Cote Water Putty" and glue+nail down a sheet of 1/4" floor ply on top of that. Would the peel-stick tiles adhere to a coat of spar varnish (more moisture resistance (and I have a gallon in the shed))?