What is the ruling on a girl dealing directly with a young man who proposed to her without her guardian's consent, and what is the ruling on him observing her all the way to her home?

If the young man is serious about marriage and you approve of him, give him your father's number or direct him to your home without prolonging the conversation. There is no harm if he follows you to find out your home address for the purpose of proposing, but beware of manipulators, and stay away from situations of suspicion and temptation. Studying in mixed-gender universities is a reprehensible act that leads to temptations, and if you are compelled to do so, observe the rulings of Islamic law regarding interaction between non-mahram (definition) men and women.