Hi Louis. Really you should not bump start the car down to that catalytic converters can be destroyed due to unburnt fuel reaching them. Jump starting can be done I myself have done it numerous times without any problems. When the cars are built at the factory they are often hooked up to a booster battery to get them going which is the ideal.
The immobiliser light can sometimes give away some clues to exactly what the problem is. With Gemz immobiliser light not coming on whatsoever that's what makes me think it's linked to the unit that controls the immobiliser the
In general though if you look on the internet or across any French car site you will find pages upon pages of immobiliser problems which is very poor really. With the technology available these days and the wealth of knowledge an immobiliser system should be made up which will stand up to the test of time in my opinion. Of course it's one of only a few things these days that for most is a dealer only job. Though maybe given this fact the motivation to make an immobiliser system that lasts isn't there. It would not be so bad if dealers were focused on good customer services but with most it's not the case.