I know Peercoin is proof-of-stake, but if I use an online wallet, will I receive any of the coins generated for my stake? Or do I have to have to download a desktop wallet?
If it is a web wallet that holds your private keys, then it is certainly possible that the service is staking your coins for you, but this may or may not be a feature that the developers of the particular web wallet have implemented. It is also possible that the web wallet creators take the stake as a payment for their service, which I hope the service would be transparent about.
If the web wallet is a watch-only wallet, then it cannot stake for you.