Sequential public key generation with secret private keys?
Yes, this is indeed possible. 

A scheme with such properties is e. g. used in Bitcoin's HD (hierarchical deterministic) wallets. The scheme was introduced in BIP32 (Bitcoin Improvement Protocol) and is explained in detail here: https://github.com/bitcoin/bips/blob/master/bip-0032.mediawiki

Regarding your example scenario: That is also possible with the system Bitcoin uses. However the key derivation path and the order in which the private keys are released is important in order to not leak future private keys in advance.