If you wish your offer to be treated seriously, it would be recommended to get pre-approved. As others have said, it may not be mandatory but your offer would be treated in a whole different light with a pre-approval letter.
based on my experience, and I am handeling many short sales in New York, a pre-approvel from a bank ( not a mortgage broker) is usually enough. we don't take deposits here. however what I usualy do is call the mortgage originator who signed off on the pre-approval to verify the information, credit score etc.... just so I feel comfortable that the information is sufficiant. proof of funds is required when the offer to purchase is a full cash offer. Hope this helps.
In California most lenders (I would say all( require pre-approval. You can pay the deposit, but you don't have to. It is a matter of agreement. Many agent require it just to make sure that buyers are serious enough. It can be deferred until the lender approval is obtained. If you decide to pay $1000 make sure that it goes to escrow. Are you dealing directly with the Listing Agent responsible for short sale or you have your agent?