How authentic is the incident cited to demonstrate the impermissibility of reviewing lessons in the mosque, and is it permissible to review lessons in mosques, and if not, what is the evidence for that?

Reviewing (lessons/knowledge) aloud in the mosque above what is needed is not appropriate. The matter becomes more severe if there are worshippers, reciters (of the Quran), or those engaged in remembrance (dhikr), because it causes them disturbance and distracts their khushu' (devotional humility). The Prophet, peace and blessings be upon him, forbade some worshippers from raising their voices over others in recitation. Reviewing knowledge in the mosque is permissible if it is done in a low voice that does not disturb anyone, and provided that the knowledge is beneficial and useful, while observing the etiquettes of the mosque. The Prophet's prohibition, peace and blessings be upon him, against buying and selling and announcing lost items in the mosque does not indicate a prohibition against learning and reviewing knowledge therein, for learning knowledge is one of the greatest acts of worship.