[Rversteeg]

I do not have any particular knowledge about the Rex Regina system, but when comparing the electrical diagram with a LH diagram you see largely the same setup. So the same rather crucial fuel pump relay feeding fuel injection and ignition. As a common point of failure I would start by checking/replacing this relay.
I am inclined not to give too much attention to the IAC fault code at this moment. It could very well be that this code has been there even before the starting problems and I can't see it being directly related to starting problems. If the throttle housing is clean, you should at least get some life out of the engine when starting as 2/3 of the air needed for idling does not flow through the IAC valve anyway. Compared to modern systems the diagnostic system is rather primitive and often more misleading than helpful.
Does the tacho needle move when starting? If it does, there is ignition and the crankshaft position sensor is working.

Edit: Apperently Dave and I were typing at the same time with similar advices. Two minds, one thought....