Can a car water pump work intermittedly?

I have a 2000 Hyundai Tiburon that occasionally runs hot. The temp goes from normal to very hot in 2-3 minutes. I have installed a new thermostat, chemically flushed the radiator, the hoses are OK, no leaks. Now, the car is occasionally making a grinding noise that may be coming from the water pump or possibly the AC compressor. My grandson has the car at college and it is difficult to tell exactly what is going on with it over the phone. Although he is not certain, he thinks it only runs hot when the air conditioning is on. Personally, I doubt the AC has anything to do with it running hot. My guess is it is the water pump, however, it seems the pump would work or not work. Can it occasionally fail and return to working after the engine cools? When the car cools and the water is replaced, it may be OK for several>weeks. So... I have three questions: 1.) Can a water pump not work sometimes? 2.) Can the cooling system (AC compressor?) cause the car to run hot? 3.) If it's not 1.) or 2.) what could it be? Your assistance would be greatly appreciated. Thanks in advance.