Is a girl born from a relationship between my father and a married woman considered a sister to the questioner, and what are the family's obligations towards this girl?

The girl resulting from an illicit relationship is not attributed to your father. Rather, she is attributed to her mother's husband as long as she was born on his marriage bed, unless the husband disavows her through li'an (oath of condemnation). The resemblance you mention is not considered proof for establishing lineage if it contradicts the marriage bed, due to the Prophet's (peace be upon him) saying: "The child belongs to the marriage bed." Furthermore, this girl is not considered your sister. However, it is not permissible for anyone to marry her as a precaution due to the existence of the resemblance.