Is it obligatory to inform the father about his daughter leaving the house without his knowledge, even if that would lead to major problems?

A Muslim woman should remain in her home and not go out unless there is a need. If your sister goes out for a necessary reason, try to fulfill that need on her behalf. If her going out is unavoidable, then you should accompany her. However, if she goes out without a need, advise her that her staying at home is better and more modest for her. If her going out becomes frequent, inform her that you will tell your father. If you suspect her conduct, then you must inform your father to safeguard her honor and reputation.