That pic shows a classic problem with the FJ40 wiring system. If the starter or the alternator does not have a good solid ground, they will attempt to pull current through any ground that is good in order to do their job. Usually through the alternator, regulator and ignition grounds. Since these are smaller 14 or 16ga wires they WILL melt.
In addition to the starter ground to frame, and the engine block to frame, make sure the alternator is well grounded to its mating bracket, and the engine. You can use a bonding jumper of say 10 ga between the body of the alternator and the engine block if you don't want to dismount and clean the alternator to mounting brackets to engine. Just I would add that bonding jumper anyway in addition to properly grounding the alternator through its mounting bracket.
You also need the proper 3 pin latching connector for the alternator to attach ground wire and the white/green wire. I sell this connector and new terminals cheap. Part #FLF3P6.3W $2.55 + $3.00 shipping. Send me a PM to order. I also sell the White/Black wire in 14, 16 and 18ga sizes by the foot, just tell me what ga and how many feet you need.
You are going to have to un-tape that harness and see if that melting goes deep into the harness. It may have melted other wires together which will cause no end to your weird problems! If it does appear to have gone deep, and damaged other wires, the proper way to fix it will require pulling the harness off the truck, completely un-taping it, replacing the damaged wires and taping it back up. I also sell the correct non-adhesive harness tape for that job. (I also stock all the proper color striped wires for your truck)
Good luck!