What is the ruling regarding a sheikh who teaches the Qur'an day and night to his students, yet only prays in the mosque on Fridays, claiming that he relies on teaching the Qur'an, not on prayer? Is he permitted to abandon prayer in the mosque due to studying or teaching?

This Imam is mistaken, as he is a role model for his students, and his statement indicates his lack of concern for prayer, which is the pillar of the religion. A servant should not rely on his deeds but rather yearn for Allah's mercy. If he prays in congregation at home, he commits no sin, though he misses out on a great فضْل (virtue/فضل). If he prays alone, he is sinful. Therefore, he should be advised kindly and informed of the importance of congregational prayer in the mosque for himself and because he is a role model for his students.