When you drive the car with the heat off, coolant will remain inside the heater core, and not mix with the hot coolant outside the core that's flowing through the system. So when you turn the heat on, that cooler antifreeze is introduced into the main system and cools things down quickly, but it'll eventually hit the same temp as the rest of the coolant. That's probably what you saw if the coolant temp dropped as dramatically as you elude to.
Many, if not all, water pumps are made to "weep" (leak out fluid) when they become faulty, so you can identify when they have gone bad more quickly, since a bad water pump can end up being a catastrophic failure for the whole engine. Check for wetness around, or possibly a small puddle under, the water pump area after driving the car.
If your thermostat is stuck closed, coolant will not be able to flow through the radiator, therefore the lower radiator hose will never get hot. It may get a touch warm, but it'll never even get close to the hotness of the upper hose.
If it's stuck open, the engine may take longer to get warmed up, yet can still eventually overheat because the system relies on pressure buildup (typically around 15 psi) to function properly. With an always open thermostat, this pressure can never be reached.
