I would test it with an ohmmeter first, but yes, if the wire is bad you can replace it. It doesn't draw much, maybe 35 watts tops, which is only about 3 amps, so you should be able to use 18 gauge or heavier wire without a worry. If you want to be extra safe go with 16 gauge.
Yes, that can be replaced as well. I would go with a16 gauge wire there, as it is actually only the ground for the two instrument panel illumination bulbs and for the voltage stabilizer, and those cannot draw more than 5 amps either because the main power wire from the key switch to the instrument cluster has a 5 amp in-line fuse in it.
Your picture didn't come through, but you are correct that the two wires coming out of the top of the transmission are for the starter safety switch. There should be two wires coming down from the bottom of the dash in that direction and it if starts without them being connected to the safety switch wires then they are most likely just connected to each other to bypass it. If it has a Select-O-Speed transmission the starter safety switch only allows the starter to turn when the gear selector is in the Park position. If t has a manual transmission (6 speed, 8 speed or 4x4 manual reversing trans), it only allows it to start when the hi/low shifter is in the neutral position, and if it has the 4x4 torque converter auto reversing transmission it only allows the starter to turn when the main shifter is in neutral.
