Does remaining at home a week before the official working hours end, based on the manager's approval and after completing the study program, constitute a dereliction of duty and the introduction of unlawful money into one's salary?

You may not be neglecting your duty if you complete the curriculum and give the students revision exercises. If the principal is authorized to permit staying at home, then there is no blame upon you, and your diligence is appreciated. For work is a trust, and Allah the Almighty said: "O you who have believed, do not betray Allah and the Messenger or betray your trusts while you know [the consequence]" (Quran 8:27). And the Prophet, peace and blessings be upon him, said: "There is no faith for one who has no trustworthiness, and no religion for one who has no commitment."