It's normal for your AC compressor to run in heat/defrost modes.
Sounds like you have a large current drain on your battery when your engine is off. To try and isolate it, remove your negative battery cable, and place a test light between the negative battery terminal, and the negative battery cable. With a significant current, the test light will be bright. Start pulling fuses one at a time to try and isolate the circuit pulling the current. When you find it, the test light will dim.