Phil, The intake manifold heater with the bad wire will get pretty hot when its used, so ether could flash and cause damage. Its operated by a position on the key, much like 'accessory' on a car. The wiring diagram will show how its hooked up. For your wiring, there are several gauges used but 14 would be plenty and is a easy to work with. 12ga, would be fine if you decide to keep it, just uses different crimp on connectors. In electrical, larger isn't a bad thing, except for bending in the big sizes. I have my harness on the bench and in between other things am building a new one. On mine, one part is under the dash and connected with plugs to the section on the tractor.
The neutral safety switch located by the shifter is designed so that when the gear lever is moved to the 'S' position, you can crank the engine to start. This prevents accidents related to the machine being started in gear an moving unexpectedly. There have been serious, if not fatal accidents related to this type of thing in machinery. On my old tractor, the switch was cut off long ago and not been used because of a problem with the shuttle, where I could not shift the gear box with the engine running. I hope that is fixed with the shuttle overhaul. On my 580CK Project thread, there are pictures of why and the repair.
A wiring diagram for your machine with the alternator would be useful in your rewire efforts. I have the generator, so mine will be different.
Have fun...