Yes, power to the compressor goes directly to the battery, but what about the interior side of the circuit which is supposed to accept a +12V ACC? It's a relayed connection. Did you jump that back to the battery as well?
If you ran both leads directly to the battery, and forgot to turn off your...
Ditto again. Use a rig runner for power and just tap a dash light circuit for illumination. It's a simple find. +12v when you turn the lights on, 0v when their off.
With the rig runner I use a 10 minute relay that turns on when it senses 13.2V and turns off when voltage drops below that...