It is not permissible for a menstruating woman to enter the mosque unless she is only passing through. However, she may sit in a place that is attached to the mosque or close to it and listen to the Khutbah.

If the birth took place without any bleeding, then the woman is not regarded as being in nifas. So, she should fast and pray like all other women who are pure. She should do ghusl so as to be on the safe side.