[ainarssems]

You should get immobiliser warning light if it is immobiliser issue if the light is working. Another way to confirm/eliminate immobiliser is - turn on ignition and start engine immediately, so the engine starts and dies within second or 2 because it had couple of seconds of runtime before immo blocked it. Second time turn on ignition, wait 5 seconds to give time for immo to block engine running and try to start, if it still starts it's not immobiliser problem.

Considering that you changed air filter before issue started I would be looking at MAF, it's wiring and unmetered air leaks, but than again it should still run at high rpm if you applied throttle, at least for cable throttle cars but as yours is drive by wire it could still be the problem.

If the tank is empty and you put in some petrol it will drain straight into fuel delivery unit so even if suction jet pumps are not working you would get more than couple of seconds running time, at least couple of miles of driving.

It could be fuel pump or relay issue as well, my friend had a very similar issue on Ford Focus, turn on ignition, fuel pump and engine runs for couple of seconds, failing pump draws too much current and relay cuts out, engine dies. At the time at Tesco car park I swapped full beam light and fuel pump relays for him as they are the same, well actually most of them are the same on Focus but I thought full beam was the kind of thing he could get by without if it was relays fault. Both full beam and fuel pump was working then and he was driving it without an issue for several month since but had to change fuel pump eventually.

If the fuel was low it could be air in fuel lines, just try turning ignition on and off for couple of seconds at the time for 10+ times for fuel pump to prime and flush out any bubbles.



Edit

Is the car on incline? That could explain weird fuel gauge readings, still should run OK if you put 5l of petrol in it. Now that i come to think about it, if somebody messed about in the fuel tank before they might have dislodged fill tube so that it drains any entering fuel in the main tank instead of fuel delivery unit in which case you will need like half a tank before it runs properly.