Car wont start unless door is open

ratty dart 340 said
There is also this bulky black wire that appears to have gotten very hot at some point.

Could be the alternator output wire. That supplies power from the alternator to everything when the engine is running. If there are a lot of electrical items running at the same time and/or the battery is recharging at high rate, any weak spote in that line can overheat.

labelled 'R6' in the drawing