Is it permissible for a student to teach another student for free in major subjects, and is that considered unfair competition to teachers who charge for their extra hours?

There is no objection to teaching beneficial worldly subjects to students, as it is an act of kindness, facilitation, and love of good, according to the Prophet, peace and blessings be upon him, who said: "Whoever among you can benefit his brother, let him do so." It is a collective obligation (fard kifaaya), and the one who performs it will be rewarded if he intends to serve Islam and Muslims.