I’m not completely clear what you mean about proof of work vs stake on Nostr but hopefully I can clarify a few things.
1. The relays are archiving your notes and other events. Since the entire event is cryptographically signed they can’t change it in any way.
2. Public key private key cryptography signs the event and proves (just like in Bitcoin) that whoever sent the event has access to the private key.
3. Whenever you make information public there is no way to stop people from saving that information. This is inherent no matter how you publish that information.
4. Notes are only tied to your public key. If your public key is tied to your physical identity than your notes are tied to you. But you don’t have to associate your pubkey and physical identity.
5. Whenever you make information public it’s difficult to control how others use it. You could try to distribute it under a license and then sue people who violate that license but no matter how you publish information it’s extremely difficult to control how it’s used. This is not unique to Nostr.