Is a room designated for prayer considered a mosque? Is it permissible for someone who is late to pray behind a follower? Is it permissible for someone who has already prayed in congregation to repeat the prayer with someone who is late, in order to help them earn the reward of congregational prayer?

A room is not considered a mosque unless it is a real mosque, so it does not take on the rulings of a mosque, such as the tahiyyat al-masjid (two rak'ahs upon entering the mosque) or the permissibility for a menstruating woman or someone in a state of major ritual impurity (junub) to stay in it.

Scholars differ on the obligation of congregational prayer in the mosque: Hanafi and Maliki schools: It is a confirmed Sunnah. Shafi'i school: It is a communal obligation (fard kifayah) (according to the soundest opinion in the school). Hanbali school: It is an individual obligation (fard ayn) for those for whom Friday prayer is obligatory, and they do not stipulate the mosque for it. Zahiri school and Ibn Taymiyyah: It must be in the mosque unless there is an excuse.

It is safer to adopt the last opinion. So, if there is a nearby mosque whose call to prayer can be heard, then the congregational prayer should be performed there. Otherwise, you are excused.