Is it permissible for me to ask a seller who offers installment plans to purchase a property he does not own, with my promise to buy it from him in installments at a pre-determined profit? If he actually purchases it, is it then valid for me to buy it from him in installments? And what is the ruling if we do not specify the profit beforehand?

This transaction falls under Murabaha (cost-plus financing) sale. It is not necessary to pre-determine the profit that the buyer will pay to the seller in installments, because what happens between them initially is a promise to purchase, not a binding contract. Therefore, there is no harm in some ambiguity or generality regarding the profit. However, when the transaction is concluded, everything must be clearly defined to prevent uncertainty and gharar (excessive ambiguity/risk).