Is the questioner considered a pimp (dayyūth) or disobedient to his parents (ʿāqq) given the family situation he described, especially with his inability to impose the hijab on his sister or expose her actions for fear of escalating matters and threatening her life?

We advise you to take it easy on yourself, for you are not a dayyuth (one who lacks jealousy for his female kin) but rather a ghayoor (one who possesses protective jealousy). You should continue to advise your sister gently and kindly, and make supplications for her. You may even shun her if you believe that would benefit her. You should gently remind your father of his responsibility towards her. Remember that advising one's parents is not an act of disobedience. You must persist in offering advice, striving for wisdom and good admonition, and be a good role model. Be sure to obey Allah and perform your prayers.