I would start by looking through the entire electrical system. Check all cables for corrosion. I would also check to make sure that there is not a substantial draw that is draining the battery dead. Also check the fuses and wiring and see if you can track down why it is overheating. If I had to guess, there is an issue somewhere in the power distribution side.
Some of the things you mention have relays that run through the main power distribution block. The High speed of the HVAC blower usually is on its own relay. Cooling fans are run by relay, etc.
At one point on my Regal, I had a loose wire going to a power distribution block, and the only way I knew about it was that the starter wouldn't engage, and there would be times the vehicle would quit running, with a wonky voltage reading. They would jump start it (applying power to the loose connection) and then it started right up. Once I found and tightened the connection down, no more problems. It happened to be the power block that had the starter relay, and some switched circuits on it.