Your domain is a predictable derivative of your hidden node's key, which is randomly generated. It's cryptographically impossible for the same domain to be created more than once.

Yup. Further, you can generate billions of keys until you find one where the public key matches some sequence you define beforehand. I brute forced one that starts with "superkuh". It only takes some tens of minutes to an hour or two on a GPU from 2010.

As long as you keep the private keys private you get traffic to that public key and "own" the domain.

Not impossible, just unlikely. Collisions can exist in theory in any hashing algorithm (though it may be a 2^2048 large address space or something)

Which is what "cryptographically impossible" means. Even with all the computation on the planet for a million years you can't get a glimmer of a chance.

You generate a public/private keypair for your service. Your domain is a short hash of your public key. Whomever owns the private key, owns the domain name.