The friends in law school should be focusing on providing you with the standard analysis -- it will help them with their exam scores!

A contract is effective when there is an offer, an acceptance, and consideration exchanged. It can also be seen as two promises (the offer and acceptance). However, both must know what the deal is and this was not a service contract but a sale of goods.
In rereading what you wrote, it seems that your understanding is that it wasn't an option contract but an acceptance of the offer you made to sell the the car. The mistake your law school friends make is in stating that you provided part of the service when the proper analysis would be that
the buyer partially performed on the contract. You made an offer to sell the car for $10,500.00, they accepted the offer, and payment terms were that they would pay $500 by check and $10,000 in a cashier's check. Therefore, the buyer may have breached the contract and would be liable for damages. Was the title signed over to the car? Not likely. You should also ask your friends about the "statute of frauds" which requires that contracts for the sale of goods for a value greater than $500 be in writing for the precise reason of what happened here -- it is difficult to determine exactly what the deal was.
Assuming the buyer breached, should damages amount to $500? I'm not sure this was a point agreed upon and my understanding of the law is that a breach of contract in this context would allow you to recover "expectation damages", the amount you expected to make as the benefit of your bargain. You still have the car to sell and if the market price dropped, you could recover the difference. You might also be able to recover other damages that could result (ask your friends about reliance damages and consequential damages) but I'm not sure that you have any here either.
Now if the other party disagrees and they take you to court, they will may say that it was a good faith check to hold the car and that they would decide over the weekend what to do. What happens if this went to small claims? I think it's likely that they'd get some of the money back as you were not truly damaged as (1) it's unsure whether there was a deal that was accepted and completed, (2) even if there were, you weren't truly damaged as a result. Regardless of these points, small claims decisions often skirt the limits of the law and awards are made of what "feels" like the moral decision... I'd think you the court might have you pay a significant amount back to the buyer.
Does that mean you don't have leverage -- you do as you have check in hand. As you are an MBA in finance, you can surely appreciate what that means. You can also appreciate the fact that when you make deals of consequence you always get the pen to paper as soon as possible! You may want to settle here and avoid the dispute and the time you may end up spending with it...