Water pump/thermostat

Just recently my girlfriend's 2000 mustang (80k) has been having heating/cooling issues. First of all, the engine occassionally overheats at a slow rate; inside the car, the heater goes back and forth from cold to hot, but it won't just blow hot air consistently. The thermostat seems to be doing its job; yet the water pump may have a leak because the area surrounding the pump is covered in coolant, but there are no obvious leaks on the pump/gasket. So i'm guessing I need to change the pump and gasket and while i'm at it i'll put in a new thermostat. However, my question is: if the water pump is going bad would that also explain the the heating issue inside the car (whereas the heater blows cold and hot air)? Thank you for the help. Any input is appreciated.