What is the ruling regarding a congregation that prays in the mosque's courtyard instead of inside due to intense heat, and is Tahiyat al-Masjid (greeting of the mosque) obligatory upon entering the courtyard?

Any spot on earth becomes a mosque as soon as we designate it for that purpose, not by construction. The mosque of the Prophet, peace and blessings be upon him, was partly built and partly unbuilt, and all of it was a mosque where prayers were multiplied and where Tahiyat al-Masjid (greeting the mosque prayer) was performed. Therefore, there is no harm in congregational prayer outside the mosque if the place is part of the mosque, and it is permissible to perform Tahiyat al-Masjid in any part of it, whether built or unbuilt.