Is it obligatory for the girl to return to her grandmother's house in Damascus despite the mistreatment she received, or should she postpone her studies for a year to stay with her sister?

The questioner should seek the reason for her grandmother's distress with her. Perhaps her presence causes her discomfort, or she has treated her poorly. She should treat her kindly and be patient with whatever comes from her. Perhaps her displeasure is without a justifiable reason, and this requires patience. She should consult her family and seek guidance from her Lord (perform Istikhara) before proceeding with any matter.