When I play Satoshi Dice from my Bitcoin-QT client, I see in the block chain that it sends my remaining BTC to another address, but I don't have that address listed in my client as a receiving address. So if I play again, will it send from that new address, and can I claim any rewards that I get?
When you spend bitcoin, you spend an entire input all at once. So if you receive a single transaction that has 3.5 BTC in the output, then later when you want to send 1.5 BTC, you have to spend the entire 3.5 BTC in a single transaction. Think of it like if someone gave you a $5 bill and you then later wanted to buy something for $2. You would have to spend the entire $5 bill all at once (you can't just rip it into 5 pieces), but you would get $3 back in change.
The client program hides most of this messiness from you. It spends the full 3.5 BTC, creating 2 outputs, one for the 1.5 BTC that you are sending to someone, and another output for the remaining 2 BTC in change that gets sent back to you. The client program chooses a new private key/address, adds it to your wallet, and sends the change back to you with this new address that is now in your wallet.
So, while you don't see this new "change" address in the list of addresses you've generated in your "Receive Coins" tab, it is an address that is in your wallet, hidden from you by the client, and the client will eventually pull coins from that address to send (quite likely generating yet another "change" address). When SatoshiDice receives coins from that address and sends your rewards back to the same address, they will still show up in your wallet.