Is it permissible to divide the house among the heirs during the father's lifetime in order to give each rightful heir their share of the inheritance? And if it is not permissible, must the father give the daughter a share of the rental income from the lower floor to appease her?

If the house belongs to your mother, then your sister has a legitimate right to her share of the inheritance. However, if it belongs to your father, she does not have the right to rent for the house as long as he is alive, except for the obligatory maintenance. Money does not take the ruling of inheritance during his lifetime. Your father is allowed to gift his children, provided there is justice among all of them, and to divide his wealth among them, provided he does not deprive anyone or diminish their right. However, this is not preferred to avoid problems, knowing that delaying the possession of a gift until after death makes it a bequest that is not permissible for an heir.