What is the ruling on a woman teaching children the Quran in the mosque when there are men capable of doing so, and is it permissible for a woman to do so when there is a need for it?

It is permissible for a woman to teach children under the age of puberty, though it is preferable for men to undertake this. If a woman needs to work, she must ensure that it does not negatively affect the children, and she must adhere to the Sharia guidelines regarding her going out, and that the teaching location is separate from men. As for teaching adult males, the general rule is prohibition. Mosques are designated for worship, so care must be taken not to allow children to cause mischief in them.