Is it forbidden for a father to refuse to be just among his children, and do the children have the right to demand their rights from their father during his lifetime, especially if his actions stir resentment in their hearts?

Money belongs to the father, and he may dispose of it as he wishes, as long as he is sane and mature. The children have no right to prevent him from disposing of it. So, if he wishes to give a gift to his wife and her children from another marriage, there is no sin upon him, and you have no right to object. He is not obliged to equalize between you and his wife in the gift; rather, equalization is only obligatory between children if he gives a gift to one of them.