What is the ruling on two separate congregational prayers – one for men and one for women – in two adjacent halls? And does the ruling differ if the second congregation coincides with the first congregation in its last rak'ah or before it? Is this ruling specific to the mosque only, or does it include other places?

There is no harm in having multiple congregations in a musalla (prayer area), and it does not take the ruling of a mosque. It is permissible for women to join the men's congregation for prayer. However, women intending to pray in a musalla specifically for them outside their homes is not from the guidance of the Salaf. A woman's prayer in her home is better for her, as the Prophet, peace and blessings be upon him, said: "And their homes are better for them." If women gather for a matter other than prayer, such as memorizing the Quran, and then the time for prayer arrives, it is permissible for them to pray in congregation.