Monastery, abbey, convent, nunnery, priory, cloister.
monasteries- the dwelling place of a community of monks
Monks and nuns do the monks' and nuns' jobs now.
You just call Monks Monks and Nuns Nuns. They don't have any other names.
nuns
Monks - monastery. Nuns - convent.
Yes, there are Buddhist nuns as well as monks. They're known as Bhikkhuni.
Monks and nuns spread Byzantine influence through their missionary work.
No
Generally speaking, monks and nuns lived in monasteries or convents, and did very much the same sorts of things other people did in terms of raising food, cleaning, cooking, and so on. They did not have families, but spent their time in prayer and devotion instead. Many monks and nuns worked at specialized jobs, and there was a wide range of these from tending the sick and making medicine, to making cheese or wine, copying books, singing in choirs, educating people, and providing inns for pilgrims and other travelers, protecting travelers on the road, and even fighting wars. There are links below.
Yes
The ordination of monks and nuns and the bi-weekly recitation of the rules for monks and nuns. Some schools of Buddhism have additional rituals.
Monks live in monasteries, and nuns live in convents or in monasteries.