How to recite Divya Prabandham, Swami Desikan’s stotras including Paduka Sahasram? It will be beneficial if GSPK provides a booklet about reciting the stotras with which days (Tithi/Nakshatram/Day) on which to recite, time (morning/evening), stars etc.

In general, stotras and Prabandham do not have any set of rules for reciting unlike Veda which has prescribed hours as some part of Veda should not be chanted at nights. During a theetu, one should not chant Veda. There are many such rules.
Stotras and Prabandam can be recited during mornings or evenings, some carry special benefits upon reciting. The month of Margazhi is called Thiruppavai month also known as Anadhyaynam where only Thiruppavai is recited and no other Prabandham pasurams are recited. Other than this there are no set of rules for this unlike Veda.