Monks and nuns, although they have given their lives to God under solemn vows, and considered "religious" nevertheless are not clergy, they fall under laity. So if you are considering a pyramid of the clergy, they are not even on it.
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
monasteries- the dwelling place of a community of monks
A religious community of nuns is a convent.
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.