Is the daughter considered sinful for taking her mother’s furniture, which her mother had donated to the needy, based on her father’s permission: "Take what is in the house, and I want nothing; leave it empty"? And is this considered an inheritance?

If the father rented the apartment, no one has the right to prevent him from living in it with his wife after your mother's death. His previous agreement to house your brother in it does not obligate him to fulfill it, unless his promise resulted in a cost to your brother and the father had no legitimate excuse for not fulfilling it. In that case, he is obligated to fulfill it or compensate your brother for the damage. The father does not have the right to prevent his unmarried daughter from living with him unless he provides her with alternative housing at his own expense. What the mother left behind, such as furniture, gold, and other items, is considered an inheritance to be divided according to Islamic law after deducting burial expenses, settling debts, and executing valid wills. The mother's will to you or your brother is not valid unless the other heirs agree, as the Prophet, peace and blessings be upon him, said: "There is no will for an heir unless the heirs permit it." It is recommended to strive to satisfy the father.