I would like to ask about menstruating woman entering the mosque. I understand that they are allowed to go there in case of need, and better stay away from the place where prayer is done but should they also, based on the hadith that the Sahaba stayed in the mosque in state of janaba after performing wudu, be in a state of wudu while entering the mosque during menstruation, or is this not obligatory on them and they can enter without wudu?

They can enter without wudu because the Prophet (ﷺ) instructed Aishah (may Allah be pleased with her) to pass him (ﷺ) something in the mosque and she said, “I am menstruating", then he (ﷺ) said, "Your menses are not in your hands". He did not order her to make wudu.