What is the ruling on a girl continuing her mixed-gender education which requires travel without a mahram, especially since her mother forces her to do so to provide income for the family after the father's death and due to the presence of a disabled brother, and what is the legitimate position regarding this matter?

It is not permissible for a girl to travel without a mahram (definition). There is no obedience to the mother in her request for her daughter to travel without a mahram to complete her studies, for there is no obedience to a creature in disobedience to the Creator. The justifications mentioned do not permit committing this forbidden act. If a mahram is available, then it is preferable to continue the studies, while observing the Islamic legal strictures. The inquirer's position is to advise and warn against mingling with the girl, such as speaking to her emotionally, being in seclusion with her, or touching her, as she is a stranger to him until the legitimate marriage contract is concluded.