Fixing headphone jack
Use a multimeter to verify which wires connect to the Left and Right headphone connections.

Tin the end of each wire - leave them long for now to make things easier for you.

Set your multimeter to the "Diode Test" range or the Ohms "X1" range if that's what your meter has.

Put the headphones on your head and touch the meter leads to any two wires.  If you hear a click out of one side, one of those 2 wires is the Left side and the other side is Common.

Now touch the meter leads to one of those first wires and another wire.  If you hear a click out of the other side, you now know which wires are for the headphones.

If you hear a click out of BOTH sides, you have identified the Left and Right wires.  It's now a simple matter to find the common wire.

As mentioned in the comments, the twisted pair is likely the mic if this is a headset with a mic mounted to it.  You are going to have to try connecting it both ways to see what the polarity is.  That said: a good first guess is that Red is (+) and Grn is (-).