Who is obligated to pay Zakat al-Fitr on behalf of my sister, who fasted with us for ten days and then completed her Ramadan fast at the home of my other sister in the same city—is it my father or my sister's husband?

Your sister's father is obligated to pay Zakat al-Fitr for her if he is responsible for her financial support, not her husband, even if her husband provided for her throughout Ramadan. This is because the obligation of Zakat is tied to whoever is responsible for financial maintenance, not to whoever voluntarily provides it.