Is profit from the Amazon Merch service permissible, considering that the site handles the printing of designs on shirts and their shipping, and is there any problem with selling a short-sleeved shirt when it is not known where the buyer will wear it?

There is no harm in designing what is placed on a shirt, whether it be a drawing or writing. This is considered a form of partnership, where you participate in the design, and Amazon participates with the shirt and its marketing. It is a condition that your share be a percentage of the price, not a fixed amount, due to the saying of Ibn Qudamah: "It is not permissible for any of the partners to have an excess of dirhams (a fixed amount)." However, if the price of the shirt is fixed and you stipulate a specific amount for yourself, then there is no harm. It is permissible to sell the design alone, provided its price is known. It is a condition that the drawings and writing be permissible, and that they not be used to assist in sin. One must also avoid women's and girls' shirts that are used for forbidden ostentatious display (tabarruj).