[Qmulus]

Yes, the alarm/immobilizer module is damaged. If you have as much function as it sounds, you probaby just have an open trace on the circuit board or damaged relay. This type of damage can generally be easily fixed. I have done about a half dozen myself. Repairs with relay replacements and circuit repair should be less than $250.

In the worst case, you can buy a good used alarm/immobilizer module and have the data transferred from the damaged module to the replacement. This cannot be done with the factory diagnostics, but rather going directly from the electronics in the module. On my car I had to do this as the corrosion was just too extensive to ensure a reliable result and I got a good deal on a used module that made it cheap enough not to bother trying to repair the original. You can also buy a new module (they are less than $400 if my memory serves me) and either have it programmed at the dealer ($$$) or do the same as a used module and have the data transferred from the original.

You DO NOT need to buy a complete used set of keys/alarm module/ECU as some recommend. That actually ends up being more complicated if you want to have those modules match the VIN of your car.