Hello Arthur
Good to see you back, you have been a busy lad!
Regards your transmission, you may be perfectly correct in your assessment.
Possible causes of stalling are given as "valve sticking", "poor engine performance" and last but not least "lock-up clutch seized"
The fact that it does not give a problem when running for a while could be down to the oil getting warmer and thinner.
With your climate control, what is happening there is an age related problem. It seems to be common in Japan, and the only known cure is to replace the unit, not too cheap of a solution.
Two of my cars were doing the same sort of madness, either bleeping all the way up to the highest level, or else down to the coolest setting. No science to it. The outside temp thing blips annoyingly as well.
I have discovered a solution, but I don't know how temporary it might be; you need to blitz the car interior with heat, and have the aircon working so that the humidity and dampness from inside the car is all soaked up.
We discovered this driving home from Oxford with Jersey Girl [good job it was winter.
] As soon as the car was warm and dry inside, the aircon started to work as normal.
In your case you have the heater bypassed, right? This means your interior is getting cold and damp. This dampness is affecting the aircon computer, and setting the logic circuits askew.
You should fix the heater core first. Then when you heat up and dry out the interior, if the aircon goes back to normal, you are laughing.
If it doesn't, you will have to find a good s/h one, or fix the one you have. I suspect the problem is caused by poor and intermittent solder connections in the circuitry. It would be worthwhile opening the unit up for a look. An electronics shop could solder up any bad contacts you identify.
Joe