Can adiyen repeat in thiruvai at 10pm the stotra patam that adiyen had learnt through GSPK? Are there any specific timings for learning stotra patam?

Stotra Patam are meant for all to practise at all times. There are no rules as such and one can learn them at nights also. That is why Swami Desikan had mercifully graced us with such Stotrams.