Really glad you like the idea of deriving the receiving address from the exporter_secret.
If the Nostr group ID doesn't change, external observers can see how frequently and at what times the group communicates, even though the ID itself is anonymous. If I remember correctly, NIP-EE just hasn't specified how to update this ID yet, but it plans to in the future.
A gift wrap event is a kind:1059 event that wraps any other event. It's a general-purpose gift wrap — the inner event can be of any kind. Think about NIP-17 DMs: external observers can only see that it's a kind:1059 event.