Is the girl deficient in her duty towards her father after being subjected to molestation by him and his refusal to financially support her, and her refusal to live with him to care for her sick mother?

What you mentioned about your father molesting you is indeed a great calamity. We advise you to be patient and strive to forget what happened, so that it does not hinder you from your religion and your worldly life. You can forget it by occupying yourself with the remembrance of Allah, acts of obedience, good deeds, and keeping company with righteous people. Treat your father as a stranger if you feel suspicious about his glances and words. Do not allow him to be alone with you, and do not stay alone with him. You must be dutiful and kind to him while being cautious of him if you are suspicious. The mother's right takes precedence over the father's right if they conflict.