What is the ruling on working in the Ministry of Education and on the salary of a female teacher who proved her residency with a false affirmation to a workplace where she does not work, knowing that she performs her job in a way that pleases Allah?

Falsifying a work permit to prove residency is forbidden (haram (definition)). One must repent from it by feeling remorse, seeking forgiveness (istighfar), and resolving not to return to it. The teacher should inquire with the employer about the residency requirement. If the intention is for her to reside in the area and perform the work as required, then there is no blame upon her. Otherwise, she must cease this action unless the employer makes an exception for her.