Can you send Bitcoins from Bitcoin-Qt without downloading the whole blockchain? If I have Bitcoins in my wallet and I have downloaded the blockchain far enough so that I can see that I have received them, can I send them without having a fully updated blockchain?
No, because Bitcoin-Qt still has no way of knowing if you actually have those bitcoins.

Imagine you had the same wallet on two separate computers one with the block chain fully downloaded and one with it only half downloaded. Now Imagine you have 10 bitcoins. On the wallet with the block chain fully downloaded you send all 10 coins to your friend. This computer now reflects that you have 0 bitcoins while the one with the half downloaded block chain still says you have 10. If Bitcoin-Qt allowed you to send those bitcoins, it would be opening the network up to double spending.