What is the way to rectify a situation with a mosque official who exhibits un-Islamic traits such as lying and nepotism, especially after attempts to advise him and communicate with those above him have been fruitless? And is it obligatory to listen to and obey him, no matter how bad his behavior becomes?

Advising one another among Muslims is important, and it should be done gently while asking Allah for guidance for the person being advised. If you have advised your brother and informed the authorities, you have exerted your effort. Continue teaching the Qur'an and do not let disagreements prevent you from cooperating in goodness, for disagreement is evil. There is no harm in apologizing; it is an act of excellence, and the brothers should convince him with the truth.