Is the money that the mother took from her grandmother considered an inheritance for the sisters, and must it be returned and divided among them, or not?

The amount paid by the grandmother to your mother is part of the inheritance and should be divided among the legal heirs according to Islamic law, unless the mother proves that the amount was a gift or a debt from which she was acquitted, or if the heirs confirm her claim. The grandmother's refusal to reclaim the amount and her statement that she will live with her daughter does not explicitly constitute a gift or a relinquishment, as it might be in exchange for a share in the house's ownership.

The default is that the amount is part of the inheritance, and the mother is not entitled to donate from it without the permission of the heirs. Whatever she has donated, she is liable for it and must pay it back to the heirs, unless they consent to the charity.