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With the temperature pegging very quickly and no warm air, I would assume you aren't likely running as hot as is indicated. If you believe it really is running hot throughout the system, then a block test would be in order to insure there are no head gasket leaks etc. Given your description I think this is highly unlikely though. It sounds more like there is either no flow (via pump or thermostat) or air trapped somewhere as a result of a leak causing problems.
I would be thinking about a stuck thermostat first, a leak/air pocket second and water pump third as most likely candidates here.I would start by taking the thermostat out and see if you have coolant flow afterward... if you do not, then you know the water pump has failed.
If you do have flow, then I would put a new thermostat in and refill the system, using a radiator funnel or similar apparatus to improve air pocket removal. Run the system open up to temperature and you should have better circulation to force any air out and get you back to operating normally. Once the air is out and you know you have good circulation (pump operating, thermostat not stuck), the heat should resume normal operation and the temperature won't behave abnormally anymore (pegging when it is unrealistic for it to be that hot).
I'm stepping out on a lunch break, but if we need to discuss this further just let me know and we can continue upon my return.