Is it permissible to violate school rules that prohibit bringing phones or taking supplementary lessons, despite the need for them?

The basic principle is to adhere to the regulations and laws set by the state or school, as long as they serve a benefit or ward off harm and do not contradict Islamic law. If there is an alternative that does not violate these regulations, then it should be resorted to, such as using a Mus'haf (physical copy of the Quran) instead of a prohibited mobile phone, or seeking help from an external teacher. The determination of whether a benefit is achieved or not falls to the specialists, not to ordinary individuals.