Is it permissible for the guardian of a child to break the shoulder of a female teacher who broke his son’s shoulder, based on what camera recordings showed of her abusing children? And what should he do?

It is not permissible for the guardian of the child to punish the teacher himself. Rather, he must refer the matter to the responsible authorities for investigation and to take the necessary measures. For retribution (Qisas) and disciplinary punishments (Ta'zirat) are carried out by the Imam or his deputy; otherwise, chaos would prevail.