Welcome to the Corolla Forum!
I hope you hang around for a while.
By what you describe, guessing of course as there's no way to diagnose a vehicle problem over the Internet, by your description it sounds like the anti-freeze/coolant is low. Look at the overflow tank in the engine bay and see if the anti-freeze/coolant is below the full mark, or even lower. If it is, this indicates a cooling system leak somewhere that a professional automotive technician will need to diagnose.
With the anti-freeze/coolant being low, it's not wise to drive the car in this condition as it can overheat causing serious and expensive engine damage. By what you describe, it sounds like the anti-freeze/coolant level is so low that it isn't reaching the heater core to provide heat in the interior of the car and after the engine has run for a while, with the level being low this can cause the anti-freeze/coolant to overheat and start to boil, which makes it expand and register on the temperature gauge. The sound of water running is probably the anti-freeze/coolant boiling and moving around inside the cooling system. If this is the case, you are very fortunate something in the cooling system didn't blow, like a radiator hose and lose all the anti-freeze/coolant.
When the engine is cold, (so as to avoid having steam blow up in your face and scald you when removing the overflow tank capl),top off the overflow tank with the proper antifreeze type, (any competent auto parts store can provide you with the correct product), and get it to a competent automotive technician for proper diagnosis before an engine worth thousands of dollars is destroyed.
I hope this is of help. Let us know how you make out. :thumbsup: