Short answers, pulled from the story.
The Sanskrit word vihara means a form of rest house, temple or monastery in ascetic traditions of India. It particularly referred to a hall that was used as a temple where monks met and some walked about.
Rock-cut architecture found in cave viharas from the 2nd-century BCE have roots in the Maurya Empire period. Cave residence tradition dates back to at least the 5th century BCE according to legends derived from oral tradition.
According to Tibetan sources five great mahaviharas stood out: Vikramashila the premier university of the era; Nalanda past its prime but still illustrious, Somapura Odantapurā and Jagaddala. Each of the five had their own seal and operated like a corporation serving as centers of learning.
The earliest known gift of immovable property for monastic purposes ever recorded in an Indian inscription is credited to Emperor Ashoka and it is a donation to the Ajivikas. This created competitive financial pressures on all traditions including the Hindu Brahmins which may have led to the development of viharas as shelters for monks.
By the second century BCE a standard plan for a vihara was established consisting of a roughly square rectangular hall off which there were a number of small cells. Rock-cut cells are often fitted with rock-cut platforms for beds and pillows while the front wall had one or more entrances and often a verandah.