Menstruating women entering rooms attached to the mosque

This depends on the intentions of the one who set up the waqf for the mosque. If he intended that the lower level should be part of the mosque, then it comes under the same ruling as a mosque, so it is not permissible for menstruating women to enter it. 

If his intention was that it should not be part of the mosque and that it should rather contain washrooms etc., then this level is not considered to be part of the mosque and does not come under the same ruling as the mosque, so it is permissible for menstruating women to enter it and sit in it. 

Shaykh Ibn Baz (may Allah have mercy on him) said: 

If the building mentioned is regarded as a mosque and the people in the upper and lower levels can hear the voice of the imam, the prayer of all of them is valid, and it is not permissible for a menstruating woman to sit in the place that is prepared for prayer on the lower level, because it is part of the mosque. The Prophet (peace and blessings of Allah be upon him) said: “I do not allow the mosque for any menstruating women or anyone who is junub (in a state of impurity following sexual activity).” 

With regard to a menstruating woman passing through the mosque for some need, whilst being careful to make sure no drops of blood fall on the floor, there is nothing wrong with that, because Allah says (interpretation of the meaning): 

“nor when you are in a state of Janaaba (i.e. in a state of sexual impurity and have not yet taken a bath), except when travelling on the road (without enough water, or just passing through a mosque), till you wash your whole body” [al-Nisa’ 4:43]

And the Prophet (peace and blessings of Allah be upon him) commanded ‘Aa’ishah to pass him a rug from the mosque, and she said that she was menstruating. The Prophet (peace and blessings of Allah be upon him) said, “Your menstruation is not in your hand.” 

But if the lower level was not built by the one who set up the waqf for the mosque, and was intended as a storage area or a place for the things mentioned in the question, then it does not come under the rulings on mosques, so it is permissible for menstruating women and people who are junub to sit there, and it is o.k. to pray there in a clean place where there are no toilets, as is the case with any other clean place where there is nothing to prevent people from praying there. But whoever prays there cannot follow the imam who is praying above him because he cannot see him or see some of the people praying behind him, and because it is not part of the mosque according to the more sound of the two scholarly opinions. 

Majmoo’ Fatawa wa Maqaalaat Mutanawwi’ah by Shaykh Ibn Baz, 10/221 

And Allah knows best.