Outside temp reading
You likely have a failed, or missing temperature probe. On my E39, this is in front of the right front wheel in the 'pod' between the wheel and front bumper. It is accessable from underneath the car - and is a black probe mounted up at an angle through the pod. Has a 2-wire clip attached. There is additional flashing mounted that partly obscures the probe location (also protects it, I suspect), but it is there.
Since the climate control uses this probe as part of the control logic, you'll want to get this fixed to allow the AC to work properly. Suspect it is less of an issue in heating mode..