Is a girl born after my parents' third divorce, and their continued cohabitation, considered my legitimate sister from my mother, and am I a forbidden (mahram) relation to her?

If a man divorces his wife three times, she becomes unlawful for him until she marries another husband. If he then has intercourse with her after the triple divorce, that is considered adultery.

A girl born from adultery is a sister from the mother's side and is considered legally forbidden (mahram) because she is certainly attributed to her mother. As for the father's side, the majority of scholars do not attribute a girl born from adultery to the fornicator. However, some scholars, such as Sheikh Al-Islam Ibn Taymiyyah, held the view that it is permissible to attribute her to the fornicator if the woman was not in a marital bed (i.e., not married).

If your father attributed her to himself, there is no objection to that, and the rulings of inheritance and mahramiyyah (being unmarriageable kin) would apply to this lineage, making you her mahram brother. In any case, your mahramiyyah to her is established from the mother's side at least.