I sent bitcoin from my BitcoinQt wallet while it was unsynced to another wallet, do I have to wait for it to sync? The wallet shows that the transaction has not been broadcasted yet with 0 of 6 confirmations. My wallet is stuck on a block right now.
Yes, you have to wait. Otherwise you would be able to pay things with money you don't have in your wallet. 

You could have spent all your bitcoins, but since your wallet is not synced you may be able to spend money you have already spent but has not been synced, thus it is not allowed to do that.