I think the black sticky stuff is the melted foam insulation around the opening edges. Check the back of the engine belt cover, do you see the foam melt as it would be on that as well.
The foam when hot and melted will have sticky black residue. I don't think the two issues are related. One think you can try is to get a R134a recharger, find the correct port to refill and see if that is any better before spending hundreds on a new compressor.
It could be a condensor or dryer unit as well. Go with cheaper items first I always say. Still doesn't explain your stalls, that could be a third problem that is related to tune-up items. when was it last tuned-up i.e. spark plugs, oil change, coolant flush, air filter?
