Short answers, pulled from the story.
Zhiyi (538-597 CE) is considered the principal founder of the Tiantai school. He was the first to systematize the school's doctrines as an original Chinese tradition, drawing on the Lotus Sutra and the philosophy of Nagarjuna. His disciple Guanding compiled Zhiyi's three main works after his death.
The Threefold Truth is Tiantai's central philosophical doctrine, developed by Zhiyi from Nagarjuna's Madhyamaka philosophy. It holds that all phenomena are empty of inherent existence, that phenomena exist provisionally through causes and conditions, and that a middle truth encompasses both: phenomena are simultaneously empty and provisionally real. Scholar Paul Swanson describes it as the central insight around which the entire Tiantai system revolves.
The shanjia (home mountain) versus shanwai (off mountain) debate was the defining controversy of the Song dynasty Tiantai revival. The shanwai faction argued that Huayan and Chan teachings were compatible with Zhiyi's original doctrine. The shanjia position, led by Siming Zhili (960-1028), insisted that the original Tiantai view was distinct and superior to the Huayan-influenced doctrine of a single pure mind. Zhili's view became the orthodox position during the Song.
The Japanese Tendai school was founded by Saichō in the ninth century after he studied under the seventh Tiantai patriarch, Daosui, in China during the late Tang period. The Korean Cheontae school was established during the twelfth century. Both branches went on to play major roles in the development of Buddhism in their respective countries.
This doctrine was advanced by the Tang patriarch Jingxi Zhanran (711-782) in his work the Jingangpi (Diamond Scalpel). Zhanran argued that because Buddha-nature is all-pervasive and synonymous with Suchness, even insentient objects such as plants, stones, and sounds possess Buddha-nature. He wrote that every blade of grass, tree, pebble, and particle of dust is perfectly endowed with Buddha-nature.
The Lotus Samadhi Rite of Repentance is the most prominent Tiantai ritual, established by Zhiyi (538-597). It combines repentance through physical activity, including prostrations, offerings, chanting, and mantra recitation, with formless repentance through contemplation of emptiness. According to Daniel Stevenson, since the end of the sixth century the Lotus repentance has been one of the mainstays of Tiantai practice and its ritual forms continue in the modern world.