The unit would require both accessory and constant power to have 12 volts for the unit to fully operate. With the key off, you accessory wire from the car, will not be energized, therefor, the radio wont work.
Your issue is as simple as unplugging the head unit and see if the battery still drains over night. If it doesn't, another test would be to program some radio stations, turn the key off for a few minutes, turn it back on and see if it kept memory. Most of the time, when you wire up constant and accessory backwards, the radio wont keep memory.
Not sure if anyone else made these points as I only skimmed through the whole thread.