FWIW, I am not a big fan of the "ultra dirt cheap" touring mentality or stealth camping (in the US that is). And normally I'd advise you do some touring first to get some experience. I kind of doubt that will happen though, so...
A) Don't skimp on fixing up the bike. Get the widest tires you can fit on the bike, a full tune-up, and replace parts as necessary.
B) Basic repair gear:
- spare tube
- spare tire (get a folding tire by the way)
- spare spokes (optional)
- multi-tool
- tire levers
- frame / mini-pump
- patch kit
- chain tool
- extra chain links or "master links"
- lubricant
C) "Trailer" vs "Pannier" depends on preference. You can likely get a BOB trailer used for the same as panniers and a rack, and that may be a better choice if your bike does not have eyelets for racks. Just keep in mind that trailers are
huge and will encourage you to drag along stuff you may not need.
D) Yes, the Adventure Cycling maps are worth it, as long as they go where you need to go.
E) Get some real cycling shorts (2-3 pairs). You can get running shirts for $15 a pop at Target (the Champion stuff). It's not as good as a $50 cycling jersey, but will be more than sufficient for your uses.
F) If you're going to stealth camp, check the state / local laws about trespassing first. I'd expect that some states are more tolerant than others in this respect.