Is paying for a car in cash and registering that in the dealership's records sufficient for possession, or is it necessary to transfer ownership via the vehicle registration (istimara) through the traffic department, given the seller's intention to sell?

If the sale is concluded and the car is taken possession of, then it is permissible to dispose of it. However, it must be moved from the place of sale before disposing of it, due to the prohibition of the Prophet (peace and blessings be upon him) against selling a commodity where it was bought until the merchants take possession of it and move it to their premises. If the two parties to the transaction separate, the sale becomes binding.