What is the ruling on purchasing a car in installments, where it is reserved from a dealership, then the dealership designates a specific bank that buys the car and gives the purchaser a check for the dealership to take the car, knowing that the bank did not own the car for a sufficient period of time?

It is permissible to purchase a car from the bank in installments if the bank buys it and it enters into its possession and guarantee before selling it to you. There is no specific period required for the commodity to be established in the bank's possession. However, if the bank pays the price on your behalf to recover it from you with interest, this is prohibited usury (riba). The consideration in contracts is based on their meanings, not merely their wordings.