What is the legitimate Sharia mechanism for establishing a family fund with returns from the father's properties, to be distributed equally among brothers and sisters, while imposing rent on sons and daughters residing in his property? And in this case, is the gift distributed with the male receiving the equivalent of two females’ shares?

It appears that your father's action is correct and perfectly just in renting out the apartments to his children. It is not a condition for the rent to be equal to that of other apartments, due to differences in specifications, as long as there is no favoritism shown to some children.

If some of the children are unable to pay the rent, it is more appropriate for the father to waive the rent for them or to give them a gift, as it is permissible to single out some children for a gift due to their need.

The preponderant opinion regarding the distribution of gifts among children is to equalize between males and females. This is the view of the three Imams, and they based their arguments on the hadith: "Would it please you that they are equally dutiful?" He said, "Yes." He said, "Then not [favoritism]." And "Equalize between your children in giving gifts, and if I were to prefer anyone, I would have preferred women over men."