I would concentrate on getting the heater fixed and forget the AC for now. Until someone puts some freon in it you won't know the problem there.

Yes there should be heat on both sides of the aux pump, it possibly has some obstruction inside it, I agree it should set a code if it isn't working. Or there may be an air lock inside that hose by the pump is maybe another possiblity. Or the pump has failed in such a way that it isn't setting a code, like the impeller is broken,

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