Short answers, pulled from the story.
Chan derives from the Sanskrit word dhyana, meaning meditation or meditative state in Buddhism. The Chinese word Chan came from an earlier form channa, which was itself a transliteration of dhyana. The same word traveled into Korean as Seon, Vietnamese as Thien, and Japanese as Zen.
Bodhidharma is credited by tradition as the founding figure of Chan in China, described in sources variously as coming from Persia, South India, or as the third son of a Brahman king. Historically, Chan developed from the 6th century CE onward during the Southern and Northern Dynasties and Sui dynasty. Little contemporary biographical information about Bodhidharma survives, and the full lineage narrative was constructed during the Tang dynasty.
The Platform Sutra is among the earliest and most widely studied Chan texts, attributed to Huineng (638-713), the sixth patriarch, and dating to at least the 9th century CE. It provides a popular account of Huineng's life, reconciles the tension between sudden and gradual enlightenment, and reflects the growing importance of the Diamond Sutra in 8th-century Chinese Buddhism. Huineng cites five major sutras within it, including the Diamond Sutra, the Lotus Sutra, and the Lankavatara Sutra.
Chan was transmitted to Korea during the late Silla period (7th through 9th centuries) and received its most significant consolidation from the Goryeo monk Jinul (1158-1210), who introduced koan practice and founded the Songgwangsa. In Japan, Zen was introduced as a separate school in the 12th century when Eisai established the Rinzai lineage; in 1215, Dogen traveled to China and later founded the Soto school, the Japanese branch of the Caodong tradition.
Gong'ans are recorded snippets of encounter dialogues between Chan masters and students, collected and commented upon in texts like the Blue Cliff Record (1125) and The Gateless Gate (1228). The practice entered its decisive phase during the lifetime of Dahui Zonggao (1089-1163), who introduced the huatou method. Gong'an practice was most associated with the Linji school, though the Caodong school used it on a more limited basis.
Sudden enlightenment holds that insight into one's true nature occurs all at once, with no misunderstanding possible afterward, a position associated with the Southern School and promoted by Shenhui in the 8th century. Gradual enlightenment holds that awakening unfolds in stages and requires sustained cultivation. The dichotomy was largely a polemical construction: Guifeng Zongmi, a fifth-generation successor to Shenhui, argued that sudden awakening into one's true nature still requires gradual cultivation to attain full Buddhahood.