Blockchain.info keeps your wallet file which contain encrypted private key/seed, but they claim they don't have access to user's wallet and it's been proved with their open-source software even though i think there's no sure way to verify if blockchain.info use exactly same software within their website/app without add any backdoor.
As for other private info they store, i think only your email address, IP/browser details & mobile number (if you add one).
If you have privacy concern, i suggest you to use Desktop wallet, whether it's SPV or full nodes wallet such as Electrum and Bitcoin Core.
Yep, that is exactly the case. There is no guarantee at all that they don't have the access to your wallet. And since there is no way to prove that, I don't trust them.
Generaly I don't trust web wallets much not just because of privacy but also security issues so I stronlgy suggest to avoid them. You always have the choice with so many different types of wallets.