It doesn't really blow it out it just warms the cold air.
No, not if it's on a heat setting. Most home thermostats have a "Fan" setting that will blow cold air if the attic is cold.
heater core possibly plugged and leaking
No! Thermostat.
The front expansion valve is clogged.
Could be a bad thermostat.
Your heater will blow cold air only.
Low coolant in the system. Defective heater control valve. Clogged heater core.
engine not warm, low coolant, or stuck thermostat
you would have to see a mechanic about that problem
One of the reasons a 2000 Jeep Cherokee heater would randomly blow cold air and then hot air and then cold might be that there is a problem with the hot/cold duct. The air or a vent might be inadvertently blowing periodically.
heater switch needs replaced
the blown head gasket would make it over heat. but it wouldn't cause the heater to blow cold, it should actually blow hot if that is wat happened