Does the print-on-demand transaction—in which the author or designer shares profits with Amazon after printing and selling the book to the buyer—fall under the prohibition of the Hadith "Do not sell what you do not possess"? And is it Islamically permissible?

There is no objection to selling books in this manner. It is a valid and binding manufacturing contract (istisna') for both parties, provided its pillars and conditions are met. These include: specifying the type, kind, quantity, and required characteristics of the manufactured item, and setting a deadline. It is permissible to defer the entire price or pay it in installments, and the contract may include a penalty clause unless there are unforeseen circumstances. There is no objection for a writer or designer to collaborate with a company such that the writer/designer provides the material or design, and the company handles the printing, production, and delivery, for instance, in exchange for a known percentage of the price for each party.