What is the ruling on listening to the Qur’an while bathing when a recorder is present in the room, and what is the ruling on playing it during prayer to soothe a child and ensure he does not cry?

There is no harm in listening to the Quran in the bathroom or elsewhere, and there is no harm in playing a Quranic recording for a child, especially if it calms him when he hears it. This is because making the remembrance of Allah heard by children is permissible, as evidenced by the Prophet, peace and blessings be upon him, performing the adhan in the ears of Hasan and Husayn when they were born.