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It most certainly could. Most new cars use a single belt to run the "accessories" like the alternator, AC compressor, power steering and...the water pump. With no belt, no coolant is flowing through the engine and into the radiator and the car will quickly overheat.
If it's a creaking noise then it is most likely that your compressor is broken.
drive it without coolant
yes
Try changing the themometer
Yes
Not likely but possible.
No. It actually must remain in the car, because it is on the serpentine belt.
Not unless you loose enough oil to cause a lack of lubrication.
no but if you try to run the car with the water pump out you could overheat your engine
Unplugging an a/c compressor shouldn't effect the heating and cooling system what so ever the only tie it has to that system is the comperessor makes the fan come on when you engage the a/c switch
Yes you need an air compressor that will give you an adjustable air suspension.