Is it permissible for the eldest sister to demand guardianship of her minor brother and to prevent her paternal half-siblings from him, and is a woman's guardianship permissible when there are men [available]?

The sister is not entitled to financial guardianship over her minor brother. The jurists have stipulated that a mother has no authority over the minor's money unless she is appointed as a guardian by the father, grandfather, or ruler. It is permissible for a woman to be a guardian by consensus. Umar ibn al-Khattab (may Allah be pleased with him) appointed his daughter Hafsah (may Allah be pleased with her) as a guardian. If the judge rules to appoint the sister as a guardian over her brother, his ruling is valid, and her siblings have no right to annul it unless the sister is unqualified to care for the child.