Short answers, pulled from the story.
Sarvastivada is a Sanskrit term meaning "the theory of all that exists." The Chinese term for the school, Shuoyiqieyou bu, translates literally as "the sect that speaks of the existence of everything." The name reflects the school's central doctrine that all dharmas - the elementary components of existence - exist in the past, present, and future.
According to K. L. Dhammajoti, the Sarvastivada school had already developed into a distinct tradition by the middle of the third century BCE, during the reign of Emperor Ashoka, whose rule lasted from around 268 to 232 BCE. Its exact founding circumstances remain debated among scholars.
The core Sarvastivada doctrine holds that all dharmas - the elementary components of existence and experience - exist in the past, present, and future simultaneously, known as the "three times." Only present dharmas possess efficacy, meaning the ability to produce effects, but past and future dharmas are equally real.
The Vaibhashika school formed the orthodox Kashmiri branch of Sarvastivada, named for their adherence to the Mahavibhasa commentary compiled at the Council of Kashmir during the reign of Kanishka II (c. 158-176 CE). They were the most systematically developed sub-school of the Sarvastivada and held the Mahavibhasa as authoritative.
Vasubandhu (c. 350-430 CE), a native of Purusapura in Gandhara, was the most important figure in the Sautrāntika sub-school and authored one of the most influential Abhidharma works ever written. His critique of Vaibhashika orthodoxy prompted twelve years of response from the master Samghabhadra. Vasubandhu later converted to the Yogacara school of Mahayana Buddhism.
In early Chinese Buddhism, the Sarvastivada Vinaya was the most common monastic code, particularly prominent in the Yangzi River area. In the early eighth century, the monk Dao'an (654-717 CE) gained support from Emperor Zhongzong of Tang, and an imperial edict required that the entire Chinese Sangha adopt the Dharmaguptaka Vinaya for ordination, ending the Sarvastivada Vinaya's prominence.