Is it permissible to sell a car by installments in the same showroom where the car was purchased, and before transferring its ownership, to a person who desires Tawarruq (monetization)? And is it permissible for the buyer to sell it to the first seller?

If the buyer is the owner of the showroom from which the car was purchased for cash, and it will be sold to him in installments, then this is a prohibited 'īnā sale, and it is as if you lent him money to take back more. However, if the buyer is someone other than the showroom owner, then this is a tawarruq transaction, which is a matter of scholarly disagreement, and the preponderant opinion is that it is permissible, provided that you do not sell the car until you possess it and remove it from the showroom, and sell it to whomever you wish, except the person who sold it to you. If the new buyer wishes to sell it to the showroom without a prior agreement from you, then this is permissible.