Hello repaint.

Also, what do I do with that wire that is soldered onto part of the trem in the trem cavity?
Your truss rod should be fine, but i'd check it again when you put it back together.

and the wire soldered to the trem is just a ground. unsolder it like you would any other wire in your guitar.
"If you can do a half-assed job of anything, you're a one-eyed man in a kingdom of the blind."
-Kurt Vonnegut
the ground wire? idk... i never took apart a strat... I left the ground in my LP and it was fine.
Quote by breakdown123
Is there such a thing as a heavy riff with out chugging on the e string?
Listen Man,

Unless you plan on taking the Fretboard off of the neck (which would require you to heat up the glue holding your fretboard to your neck. Heat it up just enough to slowly pry to fretboard off little by little. But dont heat it up not too much or you'll heat up the glue holding in your frets and inlay) I WOULD NOT RECOMEND DOING THIS UNLESS YOUR TRUSSROD IS BROKEN! That happened to me while building my Les Paul. It is very time consuming and frusteration, my fretboard has naver been the same since the surgery. If your case, just taking the strings off and the wireing out you don't even have to worry about the trussrod. SeniorBill is right though, you might want to check it when you re-assemble if there's and buzzing when you play or if your action is too high.
To actually answer the question:

The wire from the trem claw is a ground. It grounds the strings to the circuit so that when you touch them, it kills hum.

If you plan on leaving the strings off the guitar for a week or more (and you're repainting it, so yeah) loosen the truss rod all the way. It'll be putting backwards pressure on the neck if you leave it tightened, since there'd be no strings pulling in the opposite direction.