Its idling rough because you probably have a blown head-gasket and lost compression at two cylinders you need to remove all spark plugs and check each cylinders compression.
cam position sensor has failed
i would say about days sometime even more
weak fuel pump.
Probably not usually. For an idler pulley to put so much drag on the belt system that it stalls the engine, it would break or throw the belt altogether before the engine could actually stall.
No, if anything it would get cooler from having too much. -You have a different problem, possibly a coolant leak.
Yes, a busted head gasket might stall your car if the gasket causes the car to overheat. This can actually blow the heads and cripple the car completely.
Possibly the Head Gasket is blown.
It stalls even when warm
check pcv hose elbow for vacuum leak.
I would not expect a valve cover leak to cause a car to stall. A valve cover leak can fill the spark plug area with oil which can cause rough running.
air leak
air leak
If you have a loose spark plug, your car will not stall unless maybe your engine has 3 cylinders. If you have a 6 or 8 cylinder engine you would just loose some power and your engine would run a little rough.
Try cleaning the EGR valve.
Make sure you have engine coolant. If the engine isn't getting enough coolant, it will overheat, stall and possibly destroy the engine. One symptom is that no coolant is circulated into the heater. Also check the crank sensor, but check coolant first...then the crank sensor.
This could be a number of things and needs to be checked out with machines,
The heater will stop getting hot when you're low on coolant. If the coolant is low enough the engine will overheat and it will be severely damaged.