If it IS the ECU needing to relearn… check this website: https://autoguysland.com/how-long-does-it-take-for-car-ecu-to-relearn-things/ It has helpful information. 20 miles may not be enough for it to fully relearn everything. It can take a bit longer than one would think. If you still think something is definitely wrong then I'd advise taking to a dealership. It could be that your TCM needs flashed.
This is something you'll need to go under to check. Try to narrow down an area where it seems to be coming from if possible. Ball joints, exhaust mounts, basically jack that sucker up on jackstands, get underneath, n see what moves. Check the exhaust, suspension, and steering components. If it is actually under the car (not around the engine compartment) then it is probably exhaust. Alot of exhaust systems use rubber components, and these can fail and make your exhaust rattle around.
Based on what I can see the entire refrigeration system is mounted on a skid in the bottom of the unit, like you said. Looks like there's a front cover you'll have to remove, looks like there are some screws that will hold it on. Then you'll have access to the front of the skid most refrigeration units I've worked on have 2 or 3 bolts holding the skid in place. Check the front for these bolts. They'll go down or sideways through the metal plating, bolting it in. If you can access the back there may be some there as well. Unlikely, but possible. Can you post some pics of everything so I can help you better?
If the hoses going to it are both hot then it sounds like you've got coolant flowing through it. You can disconnect the hoses from the heater core and blow some air through to confirm if you have doubts. It sounds like the door isn't moving properly. The actuator might move but if something is broken (like the part where it attaches to the door) then the actuator will move without actually moving the door. Do you have a borescope you can put in there to see if the doors are moving?
One thing I have seen used many times with success is take some small diameter strands of copper wire, bend them 90° and put them in the hole. Then screw the bolt in and it will crush and deform them to provide grip. Trim off any excess and you should be good to go. Copper is soft so it deform easily to help secure the bolt, but is stronger than, say. Toothpicks or bondo
sounds like it could be one of two things. Either the climate control switches are bad, or your blend door actuator is. https://www.vwvortex.com/threads/diy-heater-air-conditiong-fix-blend-door-issue.4445527/ has what looks like the way to replace the actuator. Dont use the part number the poster supplied he got it wrong. A commenter gave the right one. As this is the more common component that fails, it is where I would look first. Especially since you said it takes so long for it to change from hot to cold. Makes me think it is having trouble or is seizing up, and driving around (vibrations) are helping free it up to move. Its also the cheaper component so it is where I would start. The control setup im seeing is about $40, as opposed to ~$15 for the actuator. From what I can tell, open it to where you can see the actuator itself (there are youtube videos if needed) and select from hot to cold. Does it move? Take the plug off and use a meter or voltage tester to see if the actuator is getting power. No...