Your wallet is "in the cloud", and that cloud is the Bitcoin blockchain!
To my knowledge no private keys are stored in the Bitcoin blockchain and my private keys are stored in the wallet.dat file when using the regular client. Here the private keys are stored on a 'seed server', though locked with a strong passphrase?
No, your private keys are not stored at all.
Your private keys are derived from your seed when you type your password, for example to sign a transaction. But they are not stored (except imported keys).
Your seed is not stored on a server; it is stored on your hard disk, and only there.