Sending bitcoins to myself If I send bitcoins from my wallet to an address I created in my wallet, will I get more than 1 confirmation eventually?  And do I have to wait for all the block to be downloaded to my wallet so I can get coins sent to me?
Yes and yes. Nobody other than you knows that you are sending to yourself since they can't see any connection between two bitcoin addresses. So it will be just like any other transaction. If you want fast confirmation you should include a fee.

And if you are using bitcoin-qt then it's always downloading blocks automatically. The latest transactions are contained in the latest blocks so it will have to download them before it realises that you have the coins. You don't have to worry about this since it is all automatic. 

Edit: To clarify the second question you can always receive coins no matter what. But to see the latest receipts in your bitcoin-qt window and to spend those coins you have to be in sync with the network.