Hi, thanks for using JustAnswer.com. It sounds like the wiring harness will need to be replaced as well as a switch or other components that may have been damaged. I did look at the cooktop breakdown and find that the wiring harness is no longer available. So, the harness will need to be repaired (if possible) and you will likely have a few hundred dollars into the other parts and labor. If the unit is more than 10 years old, I would look at replacing it. With burning wires and things turning themselves on, it sounds like you may have a pretty good load of trouble with that cooktop.
Probably not. You could order directly from the manufacturer and maybe save some time, but I always recommend buying locally. There sure are alot of choices, though!