Help! I sent coins to my bitcoin wallet before it was properly synced! I sent coins to my Bitcoin wallet not after installing it, but before it had fully synced! Then I closed and reopened it after sending the coins.

It had generated a bitcoin address to send to, but is that address "officially mine" prior to the program actually having updated and being "officially on the network"?
Yes it is, worry not.

The client can generate valid addresses without ever being connected to the internet.

When the blockchain finishes downloading, you should see the incoming transaction.