Let me get this straight...you start the car and a ground wire burns out enough to destroy the wire? Was the car flood damaged? What happens to the "temp" wire? The issue could be affecting your transmission electrical. I would have a dealer or a trusted mechanic look at it. Sounds like a fire hazard.
The heat/AC issue is probably the radio unit. On 03-04 models, it's a known issue that can affect the AC/Heat controls (they share a common circuit board). These guys can help you with that:
http://www.shortcircuitrepair.com/