What kind of flooring are you putting down? If it's tile, I'd probably reinforce the floor a bit first with a new layer of t&g subfloor before putting the tile down... which solves your problem.
If there's any musty smell, you can clean the floor with trisodium phosphate (the real stuff, not "substitute") ... but don't soak it. Just scrub it with the tsp and a scrub brush. The TSP will probably take up the glue as well... but has a significant chance of eating your plywood in the process. ;)
I think the answer depends on what "pretty mushed" means. And before you read on, here's a caveat: my sole experience with chain link fences is taking them down, not putting them up (but I've taken down quite a few :-)).
That said, my experience is that the links will collapse in on one another, not actually bend. Which means that the fencing as a whole can be straightened out again. Also, even though the posts appear untouched, you should ensure that they're plumb.
The rails are easy to replace; every home center should carry them, as well as the caps and other accessories. The important piece that you'll need is a coupling: a short section of pipe that fits snugly over the ends of two existing pieces, to join them together (in the fences that I've disassembled, this is just press fit, not welded or brazed, but I suspect a dab of epoxy would help).
Cut the existing rails with a reciprocating saw, close to the existing rails that still have caps (you could also use a hacksaw; you won't be able to get enough clearance to use a pipe cutter). Slide the new section(s) of rail through the caps, attach them to the posts, cut to length, and slide the coupling in place.
That just leaves re-attaching the fencing. I think that you can stretch it to the rail using baling wire and a pair of pliers: loop the wire through fence and rail, and twist it to take up all slack. Then attach normally.
Best Answer
The original galvanizing of the fence was in a bath of molten tin (hot dip galvanizing).
Since this is not practical for the old fence, there are cold galvanizing compounds. They are zinc rich paints.
They will not stop rust that has broken through the original coating. Rust must be removed, then coat the bare steel.
The aerosol versions would be very expensive for painting fence, but the compound is available in a brush on version. Search "Cold Galvanizing Compound for Corrosion Control" and look for liquid brush on versions.