Is she sinful for not helping her father with the expenses of 'Umrah and the house, given her intention to help her mother and her saving a sum for emergencies, knowing that her father does not desire 'Umrah?

You are not obligated to assist your father financially for him to perform Umrah, and you are not sinful for not doing so. We advise you to help your father if assisting him does not cause you harm, as this constitutes kindness and dutifulness towards him. As for a woman traveling for Umrah without a mahram, it is not permissible except for an obligatory Umrah, provided there is a safe companionship. If this Umrah is not an obligatory one, then it is mandatory for your mother to have a mahram with her.