under the dash is a terrible rats nest, and the stereo is wired in without any insulation or anything
Before you do anything else, disconnect the battery and fix or disconnect any wires that are uninsulated (unless they are ground connections). Otherwise, you could have a bunch of melted wires and connections, or even a fire.
(I have a quick battery disconnect switch for mine. I always disconnect the battery before working on the car, I don't want the starter to come on with my hands in the fan!)
I'd pull the harness out and redo it. With the starting problem, I'd also do a visual inspection of all the wires under the hood. Look for any wires that appear out of place, broken, frayed, corroded, etc. If they have aftermarket crimp-on terminals on them, these can go bad even while looking just fine (they need to be crimped and soldered).
The jumping problem could be caused by power not getting to the starter relay that sits aft of the battery. I'd pull the wire to the start relay that triggers it to see if it is getting power, if not, start tracing back from there.
My starter relay trigger comes through a switch mounted on the clutch pedal. If the pedal is not depressed, the relay doesn't get power. That switch could be bad or not hooked up right, maybe it's part of the previous owner's theft prevention system
. For an auto tranny, I think there's a "neutral switch" on the tranny that performs the same function.