Bitcoin Wallet is downloaded I think. What next? Please excuse me if I use the wrong vocab - as I am unfamiliar with this and so will probably explain myself like a foot.
Here goes. I am on a Mac and I think I have downloaded the block chain and I now have a window called Bitcoin - Wallet showing Balance: 0.00 BTC and Unconfirmed: 0.00 BTC and Number of transactions: 0.

Now what? 
1) I thought I should be getting an address or something that I need to copy and keep safe - and 
2) I should create a Password.

I don't want to close the Bitcoin-Wallet window in case I make it disappear for ever .. 
3) Should I be getting a little picture of a wallet on my desktop?

Thank you
Addresses are not private information. If you ever want to receive bitcoins you will have to give one of your addresses to the sender. 

The information that you keep private is what is known as the "private key". Each addresses has its own private key. The private key is what allows you to spend the coins sent to the corresponding address.

Usually you don't see the private key in your bitcoin-qt window. The private key is stored in a file called wallet.dat in your bitcoin-qt folder. If you decide to encrypt your wallet by setting a password it is the wallet.dat that is encrypted. So the best way to keep it safe is to a) set a password and b) routinely backup the wallet.dat file.