How to generate random addresses from public key such that they're recoverable later using pub key only
I believe you are talking about HD derived keys, as is the previous question you linked. HD wallet structure is laid out in BIP 32.

If someone has your parent extended public key, they can derive public extended child keys using it and an index/nonce. They must have the extended key (including the chain code) But they cannot derive the child private keys. Only you can do that, using the parent extended private key. So they can tell you the index, and anyone else with the index and the extended parent public key can see which public key was used, but the only one with the private key is you.

To derive the child keys you have to at least keep the master extended private key (or the seed it came from). Without that, you cannot derive any private keys.