Most dogs, and cats are adopted on the weekends, so if you were there this week. You might want to go back beforethe weekend!
You don't necessarily need a deposit. If you visit the adoption center in person, have an application on file, and see an animal you like then you can ask that they hold the animal for 24 hours. It is done as a courtesy so it's up to the adoption counselor there that day. The practice is usually reserved for instances when other family members (human or four-legged) need to meet the potential adoptee to make sure it is a good match.
Pending approval they may work with you
I don't believe so. I am pretty sure it is first come first serve.
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