You do not own a LTC-address (or BTC, or any other altcoin) on cryptsy or any other virtual currency exchange site. You can however see your balance of all crypto currencies in the 'Balance' tab. When you scroll down, you will see the LiteCoin with your current available balance and the balance that is in possible orders. If you want to have your LTC in an address, you should download an LTC-wallet app, and create a wallet in there, and then withdraw (from cryptsy) to that wallet.
And your second question: yes, cryptsy is a crypto currency exchange. So far there is no option to trade in any real currency, but I see that option is coming in a week or so, because this text is above the BTC/USD market
This market is for Verified Accounts Only. The User Verifications system will open within the next week.
Anyhow, what happens when you withdraw LTC to your own wallet, is that cryptsy looks at deposit wallets (for other people) they own. For example person
C have all a different deposit wallet, and they deposited 1, 2 and 3 LTC respectively. You had fun trading, and want to withdraw 2.5 LTC. Cryptsy sends 2.5 LTC from the deposit wallet from person
C, so there is only 0.5 LTC left in that wallet and you have (after confirmation) 2.5 LTC in your wallet.
When you trade, there is no 'real' LTC being transferred between wallets, only on withdrawl and deposits. Otherwise you cannot trade that fast (and other disadvantages, like fees and huge blocks)