yes they can ! <3 (:
The car dealership did not sign my contract, is it still binding?
yes, but you cannot sign a contract (so, you cannot buy a car from a dealer, or used car dealer without your parent).
The car dealer cannot take the car if it is paid for according to the contract. The dealer cannot change their mind and take back the car after the contract is signed and money has been exchanged.
No, not unless you sign a contract
If the car that the dealer delivers is not the car that is identified in the contract, then the DEALER is in default, not you.
Once in contract with a car dealership you are bound by law to carry out the remainder of the contract and make all payments or face criminal punishment.
The car dealership keeps the contract. However, since the car dealer gets paid by its financier to have fully paid for the car, then the financier keeps the contract and later gives it the car owner upon final payments have been completed by the owner buyer.
Dealer is selling you the car AS-IS. They are then selling you a service contract from a different company thus the dealer is not the one providing a warranty the service contract provider is the one providing a warranty.
The dealer cannot take the car back legally. It is their mistake and they will have to absorb the costs or mistakes that they made in the paperwork.
It all depends on the contract they sign
You have to look at the papers that you sign with the car dealer. Typically they make you sign an agreement to bring the car back in the event that they can not get you financed. If they did not have you sign something, then they will probably hold the note themselves and then you have to pay the car dealer. But the short answer is yes, you can not keep the car if you can not pay for it.
If you did not sign the contract, then the car does not belong to you. However, they can charge you a daily rate for having the car (as if you had rented it).