Is it permissible for menstruating women to enter the lower level of the mosque, which contains a prayer area and other facilities?

This depends on the intention of the endower (waqif). If he intended the lower floor to be a part of the mosque, then it takes the ruling of the mosque, and it is not permissible for a menstruating woman to enter it. However, if he intended that it is not part of the mosque, but rather for ablution areas and the like, then it is not considered part of the mosque, and it is permissible for a menstruating woman to enter and sit in it.