I would like to know whether the power outlet in the cargo area and the cargo area lights are switched off some time after the car is switched off.

Would the battery run down if there is something connected to the power outlet or the cargo area light is left on?

As far as I know, the power outlet switches off with the ignition (key)?

No it does not. It is on, even after the key is out. Does it have a timer and switches off afterwards? It is again on after I unlock the car.

Can someone please answer?

Do a test maybe?

If it's giving power even after the key is out, i highly doubt it's on a time delay relay, it must just give juice all the time. Install your own delay relay, get one 12V NO.