Do I owe anything to the other party after receiving their settlement?

From the facts, I cannot see how you would owe anyone anything. Defend it vigorously.

You can tell him to "put it where the sun don't shine. You owe him NOTHING.

The question is a little vague for me because I do not know who "he" is (is it the insurance adjuster or is it the other person who hit you?). I will assume it is the adjuster. It depends on why he says you owe the money. Most likely he is asking for the salvage value of the car. When a car is totaled, the insurer must pay the fair market value of the car. When they pay this, they are entitled to have what is left of your car (since they paid for it) which they can sell for the salvage value (usually about $500.00. If this is the reason the adjuster wants $500.00, you might offer to let him have the car (if it is not drivable). Technically, depending on what papers, if any were signed, you might not owe anything. If they messed up and offered full market value and did not take possession of the car, then they may have already settled and are now trying to change the deal, which they cannot do.

Can't tell from the facts you gave in your query.

Not sure what is going on here. They may be trying to pay you in full for the loss and trying to sell you the salvage price of the car if you have their money spend it and tell them to go fly a kite . (assuming this was the liability carrier and not your own carrier. If this was your own carrier you are liable for a deductible) see a lawyer if they continue to hassle you