Is it unlawful for the eldest brother not to pay the house's dues and help educate his siblings, on the pretext that his working brother's wage with the father is not commensurate with his effort?

The father's maintenance is obligatory upon the son if the father is in need. However, if he is wealthy, it is not obligatory, though spending is commendable as a way of honoring one's parents and seeking reward. Therefore, your brother is not obligated to spend on your father, as your father is financially independent. What your father pays your brother for water and electricity bills is considered a donation, and it is permissible for the father to request these amounts from his son according to his consumption.