No cool air come's out. No temperature change at

could be an air lock in the heater rad, back flush with a hose to clear.

Hi John,
Your right with your diagnostics, you have a blend door problem. If your switching to cool air and it is still hot then you also have a heat exchange unit fault, this is the component that changes the water flow from hot to cool by shutting off. If it is not operating correctly this will give you constant hot air no matter where you temperature switch is positioned.
As for the warm air while the engine is cold, sounds like the blend door is still open to atmosphere tempreture and if it is hot outside will pull that hot air into the inside of the vehicle.
Have the heat exchange valve checked, and the only way the check blend door operation is to remove the wiring from the blend doors and have the circuit tested. If the circuit is ok then the blend doors will need to be replaced.
Hope this helps

which switch did you change? maybe you changed the switch for the gauge and not the switch for the computer

Sounds like its the mixing door actuator motor. I believe its located behind your glove box, so if you remove your glove box you should be able to see it. Its a motor that actuates the mixing door to allow only hot, only cool, or a combination of the two into the vehicle. If you can get at it and the door seems to be able to move freely, you may want to check the heater control switch for voltage change using a multimeter. Im 99% sure it is one of these two culprits. Either the actuator motor or the switch. Good luck!

check your cooling level and check for coolant leak, if you don't have enough coolant it won't get hot

Everything that is heat and cooling in a vehicle is controlled by vacuum from the engine. It appears to me that your blend door vacuum line to the temp control switch is either off or cracked/broken and not making things work the way you want/expect. It's the vacuum friend.

could be the blend door motor or the door could be broken these are common

THIS PROBLEM IS CAUSED BY A FAILED PART CALLED A "AIR TEMPERATURE CONTROL BLEND DOOR ACUATOR MOTOR", IT IS LOCTED IN THE AIR DUCT AT THE BLEND DOOR APPROX CENTER OF DASH, IN MOST CASES THE DASH MUST BE REMOVED TO SERVICE THIS MOTOR..

The heater in your car depends upon a flow of hot coolant going through the heater core in order to get hot air into the car. If the heater core is partially plugged, it will restrict the flow. If you have a head gasket that is putting exhaust into the cooling system, it will eventually get into the heater core. When this happens there is no coolant in the heater core to make the air warm. If the water pump vanes have corroded away the result will be very poor coolant circulation causing the heater to not blow hot air. If your car is equipped with a heater control valve that is not functioning properly, the coolant flow will be restricted. One or a combination of these factors is likely the culprit in your case.
if the car is not overheating, the thermostat is working fine.... replace or flush the heater core multiple times to try and eliminate sediment build-up.

If the blower is blowing air just fine, but you cannot control the temperature, you need to listen to air coming from the vents while turning the temperature from hot to cold. If you do n ot hear any change in airflow, then you may a temperature door in the dash broken. this is becoming a common problem on the Equinox
Tech JK