What is the ruling on memorizing the Qur'an and Hadith by means of loud تلقين (rote learning/recitation) in the mosque between Maghrib and Isha or after Fajr, given the presence of those who are bothered by it? And is there a solution that satisfies all parties?

There is no impediment to teaching the Qur’an in the manner described (the شیخ recites and the students repeat), although it is preferable for each student to recite individually so that the شیخ can accurately correct errors. It is permissible to teach the Qur’an in the mosque, provided that raising voices does not disturb those praying or studying.