Short answers, pulled from the story.
Buddhism entered China during the Han dynasty when trade networks along the Silk Road facilitated the movement of ideas and people. The first translations of Buddhist texts began in the second century CE.
The Taishō Tripiñaka is a massive collection serving as the primary scriptural foundation for all East Asian schools. Modern scholars rely on this edition which was produced in Japan between 1924 and 1932.
East Asian Buddhism adapted values like filial piety which were absent in Indian traditions to distinguish local forms from foreign origins. Ancestor veneration became an important part of religious practice within China while the integration of Confucian ethics allowed Buddhism to take root more deeply among the population.
The Chan school is attributed to Bodhidharma and focuses on sitting meditation techniques. In Japan it is called Zen and in Korea it is known as Seon.
The Soga clan legitimized Buddhism in Japan during the Asuka period through temple construction. New sects like Shingon and Tendai emerged independently during the Heian Period before the Kamakura period saw more schools appear including Zen, Pure Land, and Nichiren traditions.