As for the wallet, you can probably find out from the OS when the software was downloaded or the wallet's data file was created.
Did you mean file metadata?
...and when you'll want to recover an address from its private key, you won't be able because it's already taken; the blockchain won't know it was you.
Such a setup can work only in a system where it will be needed to identify you as owner.
It does not need to be so strict, it even does not need to be able to identify the owner. Just a mark that there is someone who is able to sign a message for that address.
But I do not see now any practical use for that.
Additionally, it would bloat blockchain and expose public key immediately.