I have a 2003 Hyundai Elantra GT. What is wrong with it?

[itsmettt]

Friday I drove to work with no heat. It was -5 degrees here in Ohio. The gauge said the car was normal but blowing only cold air. I check my antifreeze, everything is fine. Even opened the actual thing and saw the fluid level was fine. 12 hours later I go to leave work and the heat works fine but my Check Engine light comes on shortly after. So today I take it and find out the diagnostic code...bad thermostat. So I get the thermostat changed, then my car starts to over heat. So we changed the thermostat again...and yes we refilled the AntiFreeze levels.
Not sure what is going on...still no heat...but the gauges are going into "overheat" stage...what could be wrong?
Could a bad heater core cause this problem? Please help
At Auto Zone they do a free diagnostic thing for check engine lights. They said it was the thermostat. When I opened the radiator, I could see the antifreeze is what I meant by "levels"

Oh and today it was 30 degrees outside..which is when the overheating took place

[YMEIDONTNOe]

Heater core will not cause overheating Sounds like a head gasket

[tronary]

There is no diagnostic code for a thermostat so you got ripped twice on that one. How do you know anti freeze levels (whatever that is) are ok? I would suggest you get an antifreeze tester at auto supply. I would suggest your radiator is freezing because you do not have enough % of anti freeze. 50-50 mix is not enough at -5

[tronary]

There is no diagnostic code for a thermostat so you got ripped twice on that one. How do you know anti freeze levels (whatever that is) are ok? I would suggest you get an antifreeze tester at auto supply. I would suggest your radiator is freezing because you do not have enough % of anti freeze. 50-50 mix is not enough at -5