It depends on where you bought the car. For example, California has no automatic "buyer's remorse" period in which you can simply return a car and cancel the purchase contract... once you buy it, it's yours (sellers are obligated to offer, for an additional fee, a rider on the contract that does allow you to do this, but if you don't take that option and pay the additional fee, too bad for you).
In other words... depending on the specific laws in your location, you may not be able to get your down payment returned. Contact the seller and see what they say. If they're reluctant, you might need to speak with a local attorney who can advise you as to whether you have any rights in this situation or not. But do it quickly... even in locations where this is allowed, it generally has to happen within 3 days or so of signing the contract.
"Trade in" is the term for using the equity in your current car for partial purchase (or down payment) of a different car. I think you mean a return of car. Some states have a cooling off period where a used car buyer may return a car and only pay for usage and fees. Where I am if you sign the paperwork you own it. The car can not be returned.
This can be a tricky situation. Did you give the buyer a receipt for the down payment stating it would not be returned if they backed out of the deal? If not, then you may have to give it back. Did you hold the car for them, and in the process miss a sale due to this? Here is what I would do. If you ended up selling the car anyway, I would return their down payment.
A check given to the dealership will be cashed. This is normal in business. Whether they can keep the money or have to return it is based on what your purchase agreement reads. Where I live if the purchase agreement has "Subject to financing" on it the down payment must be returned if the dealer can not get you financing and you can't get outside financing. If it does not have that in it they get to keep the money.
If you rent a car in MA you can not return it to NY. The car is insured in MA, so if you drive out of state they will track down the car and make you return it. If the car was rented from NY that is a whole different story. You can return it in NY if you rented from NY. I did not make this question up. I'm just giving you the answer. (I think this is a stupid question anyway.)
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