Is it permissible for me to display pictures of merchandise that I do not own for sale by installment or cash, and to determine its price before purchasing it, knowing that I buy it with my own money after the customer agrees, and is it permissible to take a deposit from the customer before purchasing the merchandise as a precaution against its non-return?

Dealing with a customer to sell an item you do not own is done by promising to purchase it. It is permissible to take a "earnest money deposit" ( هامش جدية) from the customer, but not a "down payment" (عربون), for fear of their backing out. It is also permissible to agree on the price of the item, whether in installments or paid in full, before purchasing it. Furthermore, it is permissible to go with the customer to the shop to identify and purchase the item, and then sell it to them. However, it is not permissible to agree on two prices (one immediate and one deferred) in the contracting session, and then have the customer choose one of them later, because this falls under the category of "two sales in one sale," which is forbidden by Islamic law.