Overheating

If the engine does not overheat at higher temperatures, you likely have straight water for coolant and it froze overnight. The heater core is the last thing where the ice will thaw. You will get plenty of air but it will be cold air.

If the heater fan only work at times, the best suspect is the fan motor, but burned contacts inside the fan speed switch and overheated terminals on the back if it are other possibilities. If the motor has worn brushes that are making intermittent contact, lightly banging on it will often get it started. Voltage measurements will help us determine why it is not running, but those tests have to be done while the problem is occurring.

If your heater system is controlled by a computer, that is also a real good suspect. They often cause intermittent problems.