No exp with A5 but too much with many A4s so with that in mind...
Immobilizer immobilizes a couple of seconds after start and the symbol flashes on the dash. In other words, car starts normally and then dies.
Don't consider anything in the fuel system until you get the starter cranking.
A4s don't have a fuel cutoff switch of any sort, and as mentioned above, fuel cutoff is not a starter cutout.
A4s don't have any crash sensor switch that would disable the starter. I doubt A5s have such a switch.
A4s with manual transmissions do have two switches on the clutch pedal and both must be adjusted properly or the starter solenoid never gets any juice. But your's is an automatic.
On A4s, the EGR valve or anti-shudder valve is not part of the starter circuit. I can't imagine why such valve positions would be part of the starter circuit.
Your shop should be able to tell you a couple basic things:
Starter has juice? (big cable to starter motor) - Answer should be yes. If no, then first place to look is the fusible links on top of the battery.
Starter solenoid gets juice when is turned to start position? (smaller cable) - Suspect answer is no. If yes, then starter has no juice, insufficient juice, or is completely shot. No means that you have isolated the part of the circuit that is not working.
If starter motor (big cable) has juice, you can ask the shop to try "jumping" the solenoid - connecting the battery positive terminal to the solenoid positive (the smaller wire), with the key in the run position. Remove the small wire from the starter and connect your jumper wire to it. Then touch the other end to the positive battery terminal. The starter should turn the engine and it should start in a few seconds. Stop touching the wire to the battery just as you would release the key from the start position. This is much improved method as compared to jamming a screwdriver across the back of the starter touching between the big cable and the little cable. Worked well on Detroit's engines, but I can't remember if it is feasible on a VW starter.
Hope this helps.