Sound like maybe you damaged the display its self and/or display cable on disassembly? If you can try an external display, that might be a good test. The fact that the machine chimes, means the digital portion of it seems happy (passed power on test).
Another potential problem is if you are not careful when reassembling the iPhone, the top stack of components on the screen can be to out of position. This can cause the phone to close as easily as it should and leave extra pressure on the display. Sounds like probably just an incomplete seating of as connecter though.
What the heck is a “screw slot”. Is the screw itself damaged? You can replace this with another fastener of the same thread and length. I have a little bucket of screws that were scavenged from old hard drives and dead computers. I have used other screws in this case, again, of the SAME THREAD and LENGTH.
Try buying the actual OEM part. I would get from a local Appliance parts dealer. In Seattle I use Apex Appliance on Aurora., Inspect it at the store and see if it looks materially different? If the wire you say is melting is larger, that could be your issue. I also wonder if the microwave itself could be cooking the switch? It’s an odd theory, but a switch doesn’t usually have high voltage, its usually just a gate for the rest of the circuit; If SOMEHOW the microwave was leaking radiation onto the switch, that could be an explanation too. wrapping the wire in foil tape might help this. Just crazy thoughts, I know.
Some of the 15” MacBook pros' need the bottom installed to boot. They have fingers that are compressed and make contact. If you are saying the indicator on the side is interfering with charging, that is surprisingly common. You can leave it disconnected if need be. Also, AVOID the cheap batteries on eBay and Amazon. They are garbage and just waste your time. Good luck.