You might have an electrical fault somewhere, but given that it goes away when you recycle the ignition maybe not. My A/C indicator light and fog lights would randomly come on when I switched on my turn signal in the past. Found out I had a worn out piece in the turn signal stalk that was allowing the high beam circuit to close without the regular lights being on first. Don't know if it was backfeeding some circuit or just messing with the logic in the BCM because it was a condition that was never intended and might not have been coded. That's why my first thought was some kind of short when you said the A/C goes funny.
Actually, maybe it makes more sense now that I think about it. I would check your reverse switch. It's probably shorting or something and sets off all the issues when you put it in reverse, but then when you take it back out and recycle the power the short isn't there anymore and the BCM resets and doesn't see the problem anymore. Here's a screenshot from the schematics. The reverse switch power is shared with the blower motor relay. If the switch hard shorts or there is some fault in the wiring somewhere I bet the BCM will kill power on that output to protect the circuit, which would turn off the blower motor as well. Not sure why the ABS light comes on, but it might be tied in there somewhere internally as well. Reset the car and the BCM doesn't see the overcurrent situation anymore, so everything is fine. It's a long chain to follow though, because the problem could be anywhere in the circuit from the power input into the switch all the way back to the reverse lights themselves. Guessing any hard short along the way is going to cause the problem, so maybe work one component at a time and see what you can find. Maybe unplug the tail light harness to try to at least take those out of the system and see what happens, then keep working your way back. Thinking you probably have a frayed wire somewhere that is rubbing against a ground.
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