What is the ruling on the relationship between the questioner's sister and her husband, and what is the validity of her divorce that she issued to avoid her ex-husband having custody of her child, knowing that she has not left the marital home and her waiting period has not ended?

This paper is not considered valid, nor does it effect a divorce. She is still under the marital bond of her husband unless he has actually divorced her. Consequently, she has no right to the custody of her child, because one of the conditions for the mother to retain custody is that she must be free from a husband. If she marries, custody transfers to her mother. Circumventing custody is not permissible.