What is the evidence that a woman should not go out except with a mahram and that her home is better for her, and is it true that the Prophet's wives used to go with him to the battlefields of jihad?

A Mahram is not required for a woman to leave her house; rather, a Mahram is required for travel. The default is for a woman to remain in her home, but she is permitted to go out for work or permissible activities when there is a need or necessity, while observing Islamic guidelines. Islamic law does not absolutely prohibit a woman from going out or working; instead, it has set conditions for this. Some of the Prophet's wives (peace be upon him) used to go out with him for Jihad.