What I've done so far:
1. Replaced the HVAC Control Module with an old one from an Escalade. Everything is the same, except Escalade starts the AC and with the ambient temp sensor connected, is saying that there's "ICE POSSIBLE".
2. Checked wires - inside air temperature sensor, ambient temp sensor.
3. Checked ambient temp sensors - both seem to work ok, I have the same values and both of them give me a correct reading on the rear view mirror.
4. I didn't check the duct temperature sensors and their wires - I don't see any difference from the HVAC if I play around with temperatures on both sides.
5. Checked the wiring to the sunload sensor.
6. Checked the sunload sensor. I only got connection on ambient light sensor and ground. The resistance was measured indoors, 15.6kΩ. I couldn't get anything from the solar sensors. I didn't check the wiring from the Body Control Module to the HVAC Control Module.
7. Checked the inside air temperature sensor resistances and this is my biggest concern. I get 2.28 kΩ @ 35°F / 2°C and 1.96 kΩ @ 62°F / 17°C. When heating the sensor with a heat gun, it goes lower. Everything seems to be working well. But I don't have any info on how big these values should be. For comparison, I measured the same sensor from a C6 Corvette, and I got 20 kΩ @ 35°F / 2°C and 13 kΩ @ 62°F / 17°C. 10x higher values with more difference.
8. I have tried unplugging the sensors one-by-one. I also took the battery of every time so the car would have fresh ideas every time.

I'm running out of ideas.

 

you probably need to reflash the escalade unit with the hummer firmware software using a tech2

 

Would you think it's still an issue with the module?

 

you probably need to reflash the escalade unit with the hummer firmware software using a tech2.

 

Did this suppose to be the same answer or there's something wrong with the interwebs?

 

Heureka!

Inside temp sensor was faulty. Replaced it with a resistor, all problems solved. I don't have a new sensor yet, but I believe the resistances should be similar to the Corvette ones.

Thank you everyone!