Is it better for a woman to stay in her home, or to study at university to become a teacher or a linguistic proofreader and offer something beneficial to society, while adhering to modesty and avoiding mixing with men?

The default is for a woman to remain in her home. If she sees a need to study in order to educate Muslim girls, she should try to convince her father. If he permits her, then praise be to Allah. If he does not permit her, then obedience to him is obligatory, because obeying him is a prescribed duty that takes precedence over studying, which is not a prescribed duty. She can look for a suitable alternative, such as distance learning or memorizing the Qur'an, to serve the community.