Is it permissible for someone who has not memorized the Quran and does not adhere to (wearing) a beard to give lessons in the mosque, especially in the West where sects and tribulations have proliferated?

It is a condition for one who presides over teaching and spreading knowledge to be qualified, and memorizing the Qur’an is one of the priorities for that. However, if he has not memorized it and wishes to teach what he masters, there is no harm. As for shaving the beard, it is forbidden and a sin, and one who persists in it is a transgressor (fasiq). The principle is not to allow a transgressor to preside unless there is an extreme necessity, such as the absence of anyone else, coupled with the people's dire need for his knowledge. In such a case, the predominant benefit necessitates permitting him.