Like has been said; change the thermostat, it's cheap and easy. Bypass the heater core to take it out of the equation though the only way it would contribute to overheating would be through coolant loss. A rotten core could also indicate corrosion in other parts of the cooling system as well. Check the water pump operation by watching the coolant flow with the radiator cap off. That's also a way to check the thermostat operation: It should circulate and stop, circulate and stop, etc.
Radiator could be plugged where you can't see it inside the tubes, but a good circulation would indicate that it's acceptable.
And always use a thermostat (except to test). An open system can overheat by not allowing the coolant to stay in the radiator long enough to cool off, and in cold weather the engine will not be able to reach operating temperature and will foul the cylinders. I would use a 180º.
