Terminated web designer will not refund deposit

What is the name of your state (only U.S. law)? California

I hired a web designer/computer repair person to duplicate my site but with a suitable shopping cart. After four weeks of doing minimal work on the site on day 1 and 2, the site had not progressed. He mentioned a family emergency, but promised to progress the site quickly but failed to deliver. I terminated his services and requested my deposit back within 7 business days. He responded weeks later that he should have my deposit within 10 days and ASAP. Since that was not definitive, I gave him the deadline of 10 business days which is due today 4/8, however he did not refund my deposit.

Legally, how long does he have to refund my deposit?

What happens if I take him to small claims court, and my deposit is refunded before the court date? I would want the costs for filing and serving along with the 2 months' hosting fees that I paid unnecessarily due to his not doing the work?

Thank youWhat is the name of your state (only U.S. law)?

Your legal position is not as clear as you want it to be. You should have sent a letter detailing why you think he was going to breach the contract. If he did not give adequate assurances, you could consider what is known as an anticipatory breach.

Here, you fired him. You can't just fire a contractor.

There is no time limit for the return. A court has not found him in breach yet. Rescission is a remedy for the court. You don't just get to decide it on your own.