Regarding the passenger door lock: it is common in some VAG models for that door and the boot to require a double press of the key fob. It's a security feature to prevent unwanted strangers jumping into the car.
The wiring damage may not be visible externally. You could have: a break in a conductor inside intact insulation, a poor contact in one of the connectors, a short circuit inside one the components, or possibly other faults I have not thought of.
I'd say the problem is electrical, so I suggest you get the battery and alternator tested professionally as a first step. It doesn't matter what voltage you measure, the battery may still be defective enough to interfere with the ignition system.
If all else fails you could weld a nut/bar/socket on the locking nut in order to remove it. Of course that will ruin the locking nut and you would need to buy a new one. Protect the wheels from weld spatter if you do try this.
I have never heard a gearbox make the sort of sound you describe, but if it is the box it is probably too late to do anything about it. Nevertheless for peace of mind it won't hurt to get the oil changed.