Installing chain link fence gate, need advice on terminating the existing fence top rail

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.

I understand that it would be difficult to dig holes for a concrete footing for your posts. Depending where you are located and the soil types, you could drive the posts in and still have a stable fence. If you are in an area with frost in the ground, I would encourage you to set them at least 3 feet deep if the soil is stable and compacted. The basic rule of thumb is to bury at least a third of the posts, but half the length is even better if you can do it.