The short run up to high revs on startup is common to many fly-by-wire throttle cars, it's perfectly normal and nothing to be concerned about. Here's what I believe is happening:
The ECU uses three different ignition maps to fuel the car during startup.
The cranking map ensures there's fuel and sparks available and that everything is ok, then it transitions to the ignition map which provides a richer mix for startup (like putting on the choke), then once the engine has fired it transitions out of the ignition map into the main running map.
Meanwhile the ECU is checking for a valid throttle position site signal from the pedal sensor so it knows the throttle is working correctly.
Until it gets this, the engine is effectively in limp mode - the throttle setting is defaulted to a value which allows the car to be driven without a working throttle pedal - so the revs go to about 2k or so. As soon as the valid throttle site signal is detected, the revs die back down as the ECU hunts for the target idle speed by adjusting the fuelling and advance. The ECU is operating in Open loop mode here, it won't be taking any notice of the Lambda sensors as the engine is still too cold.
The length of time a car spends in each map is set in the ECU programming. On my car, we have set the ECU to hold the crank map for a minimum of 3 seconds in order to start building oil pressure before the engine is allowed to fire, but many engines will exit crank mode as soon as the ECU has got a valid response from the crank sensor and the injectors. I think the logic is that people prefer their car to start immediately and aren't that worried about building oil pressure.
