Ok , you said you go through "pinball missing". You do have 3 balls in the machine? Multiball always works the same, you need to place 3 balls in the "Caster's Club". The difficulty setting only affects the drop down target that blocks the entrance of the caster's club. This drop down target is responsible for making sure you don't send a ball in the caster's club when you shouldn't or you'd be out of balls. It is also activated after you collect at least 1 jackpot and multiball ends (on normal , or is it medium difficulty). I guess on easy the drop down target won't block the caster's club after a multiball, on normal you need to hit it once and on hard you hit it twice.
The reason why
1) The wires get so hot to touch is because of that damaged connector.
2) Your game doesn't boot with the sound board is because of that damaged connector (most likely). That same connector ALSO powers the soundboard. The 5v from the power board goes to feed the CPU board, the soundboard, the fliptronic (flippers) board. They all require 5v logic, and the more boards are connected, the more power it requires from the red wires. Your wires are already too hot , adding the soundboard most likely brings the 5v too low because the bad connection in the connector is preventing the required power to go through. Keep running the game like this and you risk damaging the board more.
Now let's pretend that without the soundboard, the 5v is stable enough so that the CPU doesn't behave weirdly. If your CPU is running correctly, and you are getting issues like pinball missing, autocast not working, the very first thing do check would be the switch connectors on the CPU board. Unplugging connectors J207 and J209 on the CPU board (the one with the batteries) a few times, again to try and clean corrosion on the pins by friction could fix many issues like your multiball issues if the reel optos aren't detected properly. These are the connectors for all the switches on the playfield.
Otherwise, if it's not the connectors, the troubles you are having, pinball missing, auto cast not working, could be related to the green-brown wire that is common to the switches where the balls are (Trough switches 1,2,3 and outhole). There might be a bad contact on one of the switches and the green-brown wire doesn't make a good connection to the other switches.
I won't go deeper into details on figuring out your switch related issues until you've fixed the power. Power is always the first thing to fix/rule out. All of your current issues could be related to power. A single IC on a board could be acting up because the 5v is dirty or too low and start causing all sorts of weird issues.