Is it permissible for my 34-year-old unmarried sister to live alone in a private house, given her inability to live with her grandmother and the refusal of her stepfather's father to allow her to live with them?

First, we advise your father to house his wife's daughter with him if possible, as there is a reward in it and it protects her from tribulations and dangers. If that is difficult, then strive to find a righteous husband for her, and facilitate the marriage without exaggeration in the dowry and expenses. There is no objection to her living in an independent house if she is safe from tribulation, otherwise, it is forbidden, based on the Prophet's saying, peace and blessings be upon him: "There should be no harm or reciprocating harm."