Is it permissible for the Imam to raise his voice in intense weeping during supplication in the mosque, causing a commotion and the voices of men and women to rise in weeping, and is the loud weeping of women considered a legitimate form of supplication, or does it fall under "wailing" for which they must repent?

If weeping is out of fear of Allah, then it is praiseworthy and desirable. It is an overwhelming emotion that cannot be described as permissible or impermissible, because it is something one cannot stop. This is unlike artificial weeping, which is not permissible, because it is a clamor and raising of the voice in the mosque, and it may fall under showing off (riya'). The weeping of women mentioned is not among the forbidden wailing (niyaha). It is better for a woman to pray in her home. Worshippers should strive not to raise their voices when weeping, as many scholars consider the prayer invalid by doing so, especially if two letters become clear from the worshipper.