Arduino water pump not working and beeping
Regarding your motor just making high pitched noise (beeping) I would say that you are not supplying enough power to your motor for it to pull in the load and get rotating, you might try to first get it started by a higher power setting (PWM) and then when it's running see how far down you can set it before it stalls (stops). 

I don't belive a 12V pump would work any different.

As to the motor and L298 chip getting hot, this is sort of normal and in some cases would require som way of dissipating this heat. A heatsink might be an option. 

If you were to overheat the L298 it will shut down before being destroyed (it should).