Sounds like air in the system or improper coolant flow. Air is not the best heat transfer medium hence the heater not working while idling. When you increase the engine speed the water must be circulating into the heater core better which would give you heat. Because of the heater situation an air pocket is likely. May be a good idea to pressure test before spending countless hours and dollars chasing the issue.
The other possibility is that the water pump is not operating within specs and not circulating the water which would result it high engine temps and poor heater performance.
I would check the following:
1) verify the cooling system is properly bled
2) headgasket is not faulty (pressure test system?)
3) blockage in radiator or heater core (flush system)
4) any possible coolant leaks
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