you have to check if the ECU has Plug N play base map on it. If you use chip in your stock ECU or aftermarket with plug N play such as PFC, then the should work fine...
My guess is your car is tuned with ECU which does not have a base map, the ignition delay is caused by poor cold start tune. It is easier to tune while engine is hot and running however cold start is tricky, as by the time the tuner got the approximately right tuning, the engine would have been warm. Then to wait for the car to cool down again takes 2-3 hours, which I highly doupt the tuner will put the car back on the dyno.
This is also consistent with your idling problem, sometimes if the tune is not done properly the car will actually stalls on its own, you got to give it a bit of fuel... you won't have this problem once engine is warm no?