Can using the heater and the A/C blow out the compressor?

No. the heater is made hot from the hot in radiator and motor. the a/c is made cold from the freon and all the a/c components. I would start by checking the compressor relay and fuses then checking the control panel where the a/c on is located that you have been pushing, and the wire that's going to the compressor to make sure it's plugged in all the way and free of dirt and grease.