Don't attach the fence to the house. Aside from putting holes in your siding (not a good thing), your fence and your house will most likely move differentially. The attachment could result in a tear of the siding, which would be a bad thing.
Place the fence post close to the house, then run the fencing top and bottom supports up to within an inch or so of the siding (do leave a little gap, for that differential movement). Place the slats on the supports, allowing the slats to get just a bit closer to the siding.
You could even install a flexible membrane, similar to a product used to seal the bottom of a garage door at the floor, between the fencing and siding.
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 best way is to put two posts directly adjacent to each other - the vinyl one to the front and the wooden one just behind it, or just inside it depending how you want the front corner to look - the wooden one should also be braced to the next wooden one as it's an end-post for a tensioned fence. Putting the wood one inside will probably look better from the road, but mock it up and see what the arbiter of appearance in your household thinks.
Might also matter what critters you are trying to keep in, or out.