It is hard to diagnose remotely, but on what you are saying, I seems to be a cold issue that isn't there when it is warm.
The crank sensor usually controls ignition spark and the camshaft times the injection, you say these have been attended to or checked.
Usual cold start issues when there is fuel obviously present are sometimes the ignition components, although often this highlights more just as the engine is warming up.
One thing I would consider is whether the engine coolant temperature sensor is reading correctly, ecu side not dash gauge, this controls what we used to call the 'choke' electronically.
The idle control valve would either cause it to stall or run high when running, if once it's started and going it behaves fairly well I wouldn't look too closely at it for now.
High fuel pressure could cause it, as the engine warms up and goes into closed loop it would notice via the lambda feedback and adjust it's self but I would look at the basics like spark condition, a diy plug clean up and regap might help, then get some diagnostic equipment and look at the temperature sensors readings when the engine and car has been allowed to rest for a period and compare the readings to see if one is out causing your issues.