I had a problem like this with my original canned diablosport tune. The car would stall with the AC on if I was in heavy traffic and doing a lot of 1st gear 2nd gear neutral 1st gear etc, every time I depressed the clutch pedal the engine would dip rpm from like 800 to 400 back to 800. If it was really bad the car would just stall. Only occured with the AC on. A custom tune solved the problem. It was an AC Idle parameter setting that was too low.
On another car, I had a similar issue but it would stall in neutral or in gear with clutch disengaged. That was a fuel injector problem, I also had a occasional cylinder misfire code with that issue.
My suggestion would be to take off the cai, put on the stock intake, and flash the tune to the original tune. If the problem sticks around, you know its not the tune.
Reasons I think it is the tune: you aren't having a CEL, and because there is a parameter on some tuners to adjust the idle setting under different conditions. Because the tune is so easy to diagnose, you should do this first. Its like 15 minutes of work.
I would expect most of the other issues to occur BOTH when the car is in neutral or when the car is in gear but with the clutch disengaged.
Also, does the switch on the clutch pedal only complete a circuit to start the car, or does it serve a dual purpose of also letting the ECU know to drop to the idle RPM?
I don't know this for sure however my suspicion is that it is both. There is a parameter for Idle Drive and Idle Neutral on the SCT tuner, I would guess it does the Idle Drive setting when the car is in gear but with clutch disengaged.
Someone correct me if I am wrong please!