Does weeping invalidate the prayer if it is due to remembering external events, such as when a woman remembered the image of a murdered child during her prayer and wept, and is it permissible to weep during prostration out of خشوع (reverence/devotion)?

Crying invalidates the prayer if it is with sound and not out of the fear of Allah. As for crying without sound, it does not invalidate the prayer, whether it is out of the fear of Allah or due to a calamity. The crying of humility (khushu') in prayer is permissible and there is nothing wrong with it; rather, it is one of the characteristics of believers.