That’s really strange, but first of all very simple have you check the fuse? I don’t really know if there is a fuse, have you measure with a multimeter if there is 12 volt on the connector at the licenseplate?

 

I'll check if a fuse exists first, for them, because I don't know.
EDIT: for the fuses, this should be the schematic:
Fuses schematic
From here it doesn't seem there's a specific fuse for those lights

For the second question see my answer below

a shorted circuit will cause that, the car will shut them down automatically to protect herself from further damages! your old one should be ok! try to connect them directly to a 12 volts supply.
For a further and better diagnosis you should check your car with Diagbox, it will tell you if the harness is shorted or damaged

I have an OBDII tool which connects to my phone, I'll try that first to see if I can see some error codes.

Also, I would really like to test the cable connection, but I wasn't able to let the connectors out. I unmounted the rear plastic cover in the inner part of the luggage rack of the car, so I can clearly see the connectors to the two lights, but I wasn't able to get them out. Any advice on that? I also wasn't able to find a service book about this.

 

a shorted circuit will cause that, the car will shut them down automatically to protect herself from further damages! your old one should be ok! try to connect them directly to a 12 volts supply.
For a further and better diagnosis you should check your car with Diagbox, it will tell you if the harness is shorted or damaged

 

the OBDII scanner elm327 or any similar scanner will not read the Body codes. they only reads the power train generic errors and some special codes.
you need to check it with Diagbox.