Is a place where the five daily prayers are performed but Jumu'ah prayer is not performed considered a mosque, having the rulings of a mosque (such as praying Tahiyyat al-Masjid), or is it merely a prayer area (musalla)? And what is the difference between a Jami' mosque, a mosque, and a musalla?

There is no difference between a mosque where Friday prayer is performed and one where it is not, regarding the legitimacy of performing the two rak'ahs of greeting the mosque (Tahiyat al-Masjid), due to the generality of the Prophet's saying: "When any of you enters the mosque, let him not sit down until he prays two rak'ahs." A mosque (Masjid) is a place of prostration, and legally, it is "any place prepared for Muslims to perform the five daily prayers in congregation." This includes places where Friday prayer is performed and places where only the five daily prayers are performed. As for a prayer room (Musalla), it is more general than a mosque; usually, its land is not Waqf (endowed), and not all five daily prayers are performed there. It does not take the ruling of a mosque regarding the prohibition of a menstruating woman or one in a state of major ritual impurity (junub) staying in it, or for I'tikaf (seclusion for worship), or the necessity of greeting the mosque. Ibn Uthaymeen clarified that prayer rooms in government departments or girls' schools do not have the ruling of a mosque.