It sounds like you're boiling the coolant which would be why the heater merely puts out warm air when the temp gauge goes to too hot.
Things to check
- is the coolant actually topped off? After running too hot you will probably find some boiled out.
- is your radiator cap in good shape? If it is worn, cracked or otherwise not sealing or holding pressure this can cause the water to boil, overheating the engine. Air bubbles are much less efficient than coolant at removing heat.
- how long has the pump been on the car?
when your looking at the still coolant the engine is at normal operating temperature? if it's not, the thermostat will be closed and not let coolant flow through the radiator. If you do not see flow or much flow without revving at operating temp then I would look into replacing the water pump.
https://www.youtube.com/watch?v=QTVPF1WdtnI watch at 2:40 and see if you have seepage too. I still think it is strange that it will work then basically stop.
Well that shoots that easy fix in the foot.
Yes getting hot water to your heater core inside the car would mean the pump is functioning and pumping water. The sudden cooling down is really strange that is why I thought of boiling, because water pumps don't just shut off unless they are electric or you broke a belt.
Has it actually gotten hot at all? has it boiled over? Go get, borrow, buy, rent, or loot an infrared temp gun. let the car get hot have someone inside to tell you when it starts overheating. Point it at the upper radiator hose and see what happens the entire time the car heats up. We might find that your thermostat opens lets all the cold water rush into the engine then close suddenly. This would be why the heater core cools down. This would also mean the issue might be the gauge or the computer cannot cope with the sudden temp change and goes high rather than low.
you should also point it at the heater hoses going into the firewall.
We are looking for a sudden rise in temperature as the thermostat opens letting coolant flow through the upper radiator hose. This should show a slow crawl upwards then the temp gun should read a rapid climb to about 100 deg c as it opens I think it will then stop or cool off from there.
The other place to aim it is near the block on the engine side of the thermostat housing if you can. This reading will show us what temp the thermostat is being bathed in. below is a pic of your thermostat housing. we want to hit just about 1/2 inch behind that gasket away from the hose side. maybe even the smaller black rubber hose going to it in that image near the "thermostat housing" arrow.
If it is cold where you live this would exacerbate the problem since a closed thermostat would let the coolant in the radiator get extra cold. This could potentially cause engine damage in the form of warped heads cracked block or heads and other potential problems. It would be the same as dumping a hot 100 deg c pan into icy cool water. To help avoid this put some cardboard in front of the radiator leaving just enough open radiator to keep the thermostat open while driving but not too little to cause real over heating.