What are the rights and duties of a divorced woman? And is it permissible for her to go out with her children for recreation, or a social occasion, or to run daily errands after sunset?

A woman in her 'iddah (waiting period) is permitted to leave her home during the day or night if there is a necessity. However, she is not allowed to spend the night outside her home unless there is an absolute dire necessity. Her going out for recreation, or for her children's recreation, or attending a social event, are not considered among these necessities.