Does the mother deserve financial support from her father, and what is the legitimate Islamic ruling on her current situation as a woman separated from her husband without a divorce, especially with the husband's claim that she is "nashiz" (rebellious)?

If a mother abstains from her husband and disobeys his commands, she is disobedient (nashiz) and has no right to maintenance until she agrees to and obeys him. She must repent and seek to appease her husband. It is not permissible for her to seek a divorce or the divorce of her co-wife if the husband is capable of polygamy, and the absence of a separate house does not prevent him from his right to multiplicity [of wives]. Efforts should be made to reconcile the parents, and the father has the right to seek assistance for his personal expenses, including the rent of the house they live in.