Is it obligatory to make up for the missed fasts on behalf of a deceased mother when the number of days she owed is unknown, and should an expiation be paid on her behalf, or should her son fast for her, and is it permissible to give the expiation to her needy sister?

If you are not certain that your mother was obligated to make up a missed obligatory fast, then you are not required to fast on her behalf, nor to offer expiation (kaffarah), nor to feed the poor. However, if you know that she was obligated to make up a fast, such as from Ramadan, then it is recommended for you to fast on her behalf, according to the preponderant opinion, and you are not required to offer expiation or to feed the poor.