What was the condition of the panel bars where the breaker is attached? A poor contact there or at any other point could cause all of the problems you describe. A poor contact would increase operating costs, but only to a small degree. The wasted energy would be heat and if significant, would show up as smoke, smell, and possibly fire. After careful reading of your post, I think a good electrician is needed at your house.

Where did the wire pop out from when you were replacing the breaker? The old breaker?

Is this device a heat pump? Rather than electrical, it sounds like you have a problem with AC unit, perhaps a bad compressor bearing. Even at 38 degrees outside, the AC compressor should still run if the thermostat is calling for cold air.

Rather than calling an electrician, I'd call an HVAC professional.