Is it permissible for a barren sister to seek Khulʿ from her husband, while relinquishing all her rights, due to her desire to care for her deceased sister's children, and is she required to observe an ʿIddah after Khulʿ?

A wife is not permitted to seek Khul' (dissolution of marriage) unless there is a valid reason, such as her detestation for her husband and her fear of not being able to uphold Allah's bounds. In such a case, she may request a divorce or seek to ransom herself from him. As for the 'iddah (waiting period) of a woman who has undergone Khul', it is the same as that of a divorced woman, which is three menstrual cycles, based on the Almighty's saying: "Divorced women remain in waiting for three periods."