Is it permissible to give the food expiation (kaffarat al-ta'am) to a poor sister, and is it permissible for the mother to give her zakat to the questioner's sister from the father's side?

It is permissible to give the expiation (kaffarah) to your poor sister; in fact, she has a greater right to it. It is also permissible for your mother to give her zakat to her, as long as she is poor. Zakat and kaffarah are not to be given to those whose financial support is obligatory upon you. You are obligated to learn the rulings of Islamic law by listening to lectures, reading books, following beneficial websites, and asking scholars.