Apologies if this is not the appropriate board for posting this, but have been all over and have not yet found a solution.
I am looking for someone to help me allow the following in my store:
From what you mentioned, I think you need to use 2 separated apps, 1 for Pre-order and 1 for Installment payment.
First, for Pre-order, you can consider using Vitals (with 40+ other features including Pre-order) or Pre-Order Now app - you can manage which product is available for pre-order and add shipping info on your product page, both apps offer that features.
I don't think there is any app can do all of what you listed, but combine this 2 kind of apps might help you sort things out.
Hope this helps.