I can def hear my pump prime on drivers door opening
(in fact its a common gotcha what NOT to do when fitting a fuel pump when its apart!)
asides from that, what they are telling you doesnt quite ring true.
Duty cycle is detemined by the ECU, what they may have done is reset the adaptation values but not duty cycle (AFAIK that isnt possible its calculated on the fly by the ECU).
What it may point to though is something like a leaky injector thats dropping the fuel rail pressure when the engines off, so that when you come to restart, it all needs to get up to pressure again before it runs smoothly.
And why it needs a few days to 'leak'
would suggest getting someone close to you using VAGCOM to measure rail pressure at idle, then if possible 1/2/3 days after (to see how far its dropped)
you could also log duty cycle for the injectors (all) during idle and running to see if one of them is 'different'
does the car run smooth and hard to the redline?
you 'could' consider injector cleaner
you should also consider wider issues and get a full diagnostic like compression test if nothing obvious comes up (e.g. head gasket etc)
odd judder at idle can be normal depending on your engine mounts and if your coil packs are OK etc