That connector shouldn't melt... it's low voltage. You can still find replacement connectors though, and that might not be a bad idea.

 

Update: I've finally fixed the issue!
So for anyone who might come across this thread in the future with a similar problem, It actually turned out to be a combination of problems:

1. There was no 5v feed to the TPS. This turned out to be caused by two connectors being mixed up. There are two connectors on the passenger side of the engine, one feeds the engine sensors and the other one goes to rear/transmission. The two actually do fit into each other. This caused the TPS (and other sensors I assume) to get no power (this also explains why I kept popping fuses for the backup light sensor)
2. I found a bad ground on the firewall. I believe this is the ground for some of the sensors (I didn't follow it, but it goes to the same part of the loom). The ground connector was fixed into place on top of the painted firewall so there was little to no ground.
3. After fixing the above things I started the car and noticed that the timing was just horrible (it sounded like a diesel engine). I checked the timing but it showed almost exactly 10° BTDC, which made no sense to me. I played a bit with the timing but couldn't get it right, then I tried moving it all the way back in the other direction and that's when I noticed the second timing mark. I turned out that someone painted the mark on the harmonic balancer in the wrong place. Once I set the other mark to 10° BTDC the engine was running real good.

I've driven it around a bit and so far I have had no issues. There's no stumbling and the throttle response is muuuuch better. It's still not a racecar, but it does actually feel like i'm driving a V8 now.