when said person accepts my offer, is there a new box where I can put contact info and my BTC address?
It's none of that person's business who you are. And if your order is filled by many different suppliers, you wouldn't want to have to arrange details with all of them.
The way an exchange work is you deposit funds of some kind. You can then place offers backed by the funds you've already deposited. When two offers match, the exchange adjusts the balances of both parties. When you're finished trading with some funds, you can withdraw them.
So you put in dollars, you can exchange them for bitcoins by placing a 'buy' order. As your order is filled, your dollar balance on the exchange will drop and your bitcoin balance will increase. Whenever you need to, you can withdraw either bitcoins or dollars from your account.
