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.
The sync of you wallet do not cause any trouble for retriving bitcoins. If somebody send you some bitcoins to your address and the transaction have been accepted you will receive you bitcoins when your client syncs.
The bitcoin client creates both a valid address and private key once it is launched. When it becomes fully synced your transaction will show up (or more specifically synced to at least the block that the bitcoin tx was initiated).
Your wallet can be left offline indefinitely, the private key that was generated will always hold that bitcoin until the wallet.dat is synced and the bitcoin is then spent.
Your btc balance is perfectly safe.
You have access to the private key to you shouldn't worry; you will recieve the bitcoin.
