Too little coolant, or a plugged coolant line.
The engine is overheating.
catalytic converter
A faulty temperature sensor can cause the ECm to think your engine is overheating, this turning the heat light on constantly. You should make sure that your engine isn't in fact overheating though.
Sounds like the ignition module located under thecoil pack is breaking down from heat.
Sounds like you have a coolant leak. When cold, put a pressure tester (hand pump with fitting to the radiator neck) and see if you can see any coolant leaking. Once found, fix the leak.
Probably a bad thermostat.
The fan should turn on whenever the engine temperature reaches a certain point. If the fan does NOT come on, the cooling system is not able to dissipate the excess heat. Yes, it can cause the engine to overheat.
Yes, if the radiator fan is not working, it can cause overheating of the engine. The radiator fan helps to cool down the engine by pulling air through the radiator to remove heat. Without the fan, the engine may not receive adequate cooling, leading to overheating.
No heat coming from the heater and the car overheating is usually the thermostat stuck closed. The thermostat is supposed to open up at a certain temperature allowing the engine to warm up quickly.
Overheating if stuck closed, no engine heat if stuck open.
no , the heater core provides heat to the interior for the passengers
There is something wrong with your radiator that it is not cooling your engine coolant. The heater has a little rad in it that sends heat to the cabin and thereby cooling the engine coolant also. If you stop the heat then your engine returns to overheating. Get this fixed or engine replacement will be in your future.