What is the ruling on using air conditioners in mosques, and are there any regulations for their use? And what is the ruling on one who misses congregational prayer because of them?

Averting harm from worshippers is required. The Sharia (definition) has made some allowances in certain rulings for this purpose, such as delaying prayer during extreme heat (al-ibrad). Air conditioning mosques is desirable for the same goal, which is to achieve khushu' (devotional humility). This should be done in a way that provides benefit without harm, taking into account the balance between benefits and harms. The assessment of these matters falls to those in charge of the mosque. The principle in such matters is what is suitable for the majority of people. Whoever is harmed by air conditioners due to illness and cannot find a place in the mosque where they are not affected, they are excused for not coming to the mosque.