Yes, that's exactly how the NUKI works.

Yes you can, I also use it with a Kaba key. Please note that a key is required which will remain permanently in the lock on the inside of the door. This means that you can only open the door from the outside with the Nuki, unless your lock supports the emergency and danger function / priority function.

Yes, nothing else is needed! When ordering, pay attention to the designation for Swiss lock (or similar) not for Euro cylinder.