Check your electrical system. You probably have a short circuit, melted switch, or ghost(and no I do not mean a creepy spooky ghost) in the system.
It is not supposed to be on when engine is running unless there is a problem.
Try replacing your Oxygen sensors they are located on the catalytic converter. I had the same problem with my 94 Caprice and that turned out to be the problem.
If the fan keeps running it's because it is electric and wired through a temperature switch not the ignition. Electric Fan running is independent of engine running. When the temperature is high enough to start the fan, it goes on. when the temp drops the fan goes off. Temp does not drop right away when the car is turned off. It can actually rise several degrees when the engine is turned off. The fan running after engine shut off is actually a good thing that may prevent radiator overflow
The compressor is probably froze up.
If the engine is running, this is normal. If the engine is turned off and the pump continues to run all the time there is a problem, possibly a faulty fuel pump relay.
Its not your engine. Its the fan running to cool the engine down. They all do it and nothing is wrong with your car.
The key has to be turned to the "Run" position, but the engine should not be running. The key has to be on to power up the ECU.
It's called dieseling, and i had the problem in my 87 Caprice for a while. A quick solution is to shut your truck off while it's still in gear and put it in park when the engine completely shuts off.
what kind of fluid, and what's a corsa?
When a check engine light comes on, it means that there are issues with the engine. It gets turned off when the person resolves the problem that caused it to turn on.
Of course your clutch and/or gearbox are faulty...
Has it always been like this or not? If it is, then it is not really a problem. It is how the stereo was installed. An stereo can be installed to be turned on whether the engine is on or off. If the stereo cannot be turned off at all, then you have a problem with your stereo.