Short answers, pulled from the story.
The Japanese monk Saichō officially established the Tendai school in 806. Saichō traveled to China in 804, studied under the Tiantai patriarch Dàosuì at Tiantai Mountain, and returned to Japan by the sixth month of 805 with texts and initiations in Chan, Precepts, and Chinese Esoteric Buddhism. Emperor Kanmu granted him recognition for an independent Tendai order.
The Tendai school has been based on Mount Hiei since its founding, with Enryakuji serving as its head temple and primary training center. Mount Hiei sits at the northeast of Kyoto, a location considered auspicious as the city's spiritual protector.
All six major founders of New Kamakura Buddhism initially trained as Tendai monks: Nichiren, Hōnen, Ippen, Shinran, Eisai, and Dōgen. Hōnen founded the Jōdo-shū Pure Land school, Shinran founded the related Jōdo Shinshū, and Nichiren founded Nichiren-shū, while Eisai and Dōgen brought Zen traditions to Japan.
The warlord Oda Nobunaga destroyed the Tendai headquarters on Mount Hiei in 1571, killing many of its inhabitants. Nobunaga's aim was to break the military and political power of Tendai institutions. Mount Hiei was rapidly rebuilt with the backing of Toyotomi Hideyoshi and the Tokugawa shoguns.
Hongaku is the Tendai doctrine of original or innate enlightenment, holding that all beings are already enlightened from the beginning rather than becoming enlightened through long practice. The medieval Tendai form of the teaching extends this to all phenomena, including grasses, trees, mountains, and rivers. It developed within the tradition from the cloistered rule era through the Edo period.
Tendai has established a presence in Western countries, though it is less widespread than Zen or Shin Buddhism. The Tendai Mission of Hawaii Betsuin was founded before World War Two and received its first bishop, Ryokan Ara, in the 1970s. The Tendai Buddhist Institute in Canaan, New York, founded by Abbot Monshin Paul Naamon and Rev. Shumon Tamami Naamon, became the first Tendai center authorized to train and ordain priests in North America.