For what it's worth, I have always seen a little engine oil in the breather hose connected to the air filter housing.
IMHO, it's normal and your stalling has got nothing to do with the oil appearing in the breather hose.
I had a stalling problem a long while back which was cured by swapping out a Fuel Pump Relay that had gone bad.
The car would intermittently stall out when slowing down to take a corner or when braking at a stop light.
Seems to me, in your car, maybe the idle valve/airflow plate is sticking.Did you try cleaning them again?