Short answers, pulled from the story.
The Japanese word tera is cognate with Middle Korean Tiel, the Jurchen Taira, and reconstructed Old Chinese dɘiaʁ, all meaning Buddhist monastery. Scholars trace the root to the Aramaic dairā, meaning "to live together," likely transmitted by early Central Asian scripture translators such as An Shigao and Lokaksema.
Buddhist temples and Shinto shrines share many architectural features and were historically intertwined as mixed complexes. A Buddhist temple is also a monastery, whereas shrines enshrine kami. The clear legal separation between the two was imposed by the shinbutsu bunri law of 1868, which forced many mixed complexes to choose a single identity.
The four main styles are Wayo, developed by the Tendai and Shingon sects during the Heian period and characterized by simplicity and natural materials; Daibutsuyō, a monumental style introduced by priest Chōgen based on Song dynasty architecture; Zenshūyo, associated with the Zen sect and featuring earthen floors, curved mokoshi roofs, and cusped katōmado windows; and Setchūyo, a Muromachi-period fusion of the three earlier styles.
All Buddhist temples carry a sangō, or mountain name, as the first component of their full title, a tradition rooted in the early practice of building monasteries in remote highland locations. Even temples in valleys or on flat ground carry this honorific, and the founding of a temple is still referred to with a term that carries the mountain meaning.
Before 1868, Tsurugaoka Hachiman-gu was both a Shinto shrine and one of the oldest Buddhist temples in Kamakura. After the shinbutsu bunri law mandated separation, it was reclassified as a Shinto shrine. Its giant Nio guardian statues were sold to Jufuku-ji, and Buddhism-related structures including its tahoto pagoda, its mido, and its shichido garan were demolished.
The four great temples of the 7th century are Asukadera, founded by Soga no Umako as the first full-scale temple of the Asuka period; Kudara Odera, which featured a nine-story pagoda; Kawaradera, notable for its asymmetrical plan with two golden halls; and Yakushiji, constructed in Nara in the early 8th century and still standing in its original layout today.