The engine firing then dying sounds more like immobiliser aerial or do you have two keys on your key ring for the car, if so try removing one of them to avoid confusing the car.
The rest of your post with idle fluctuations from 500 to 2000 RPM does indeed sound like a shot ETM, the one on your car can be repaired by one of the forum members for a little over £100 which is the cheapest option - Mike
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