Is it permissible for my father to combine my grandmother's pension, given that she suffers from Alzheimer's, with the family's income to spend on everyone, or must he calculate what my grandmother spends and save the rest for her as an inheritance to be distributed after her death?

If your father is the guardian of your grandmother regarding her wealth, then he must manage it in a way that serves her best interest. He should spend on her in a reasonable manner and preserve any surplus beyond her needs. He is not permitted to spend from it on his own family and children. Rather, he must safeguard it so that when she passes away, it becomes an inheritance to be divided among her heirs.