Not if they designed the algorithm right.

Probably not to get the key, but you could certainly use it adversarially to remove the watermark.

Removal may come down to changing every third token to a different one.

Yea, they are going to have to monitor how often similar writing pieces are being submitted, or limit the access to the checker somehow.

Europe trusts its institutions way too much. It may well be that only the police can access the checker.