Short answers, pulled from the story.
The Mahāyāna Mahāparinirvāṇa Sūtra is an influential Mahayana Buddhist scripture that uses the setting of the Buddha's final nirvana to teach that the Buddha is eternal and undying, without beginning or end. It also presents the doctrine of buddha-nature, the idea that all sentient beings possess an innate capacity for Buddhahood described as a hidden treasury within all living beings.
The core portion of the sutra was likely compiled during or prior to the 2nd century CE, with Stephen Hodge estimating a composition period between 100 CE and 220 CE based on textual evidence. The earliest form may have originated in South India during the time of the Śātavāhana dynasty, possibly in Andhra or the Deccan.
According to the Nirvana Sutra, buddha-nature (buddha-dhātu or tathāgatagarbha) is the fundamental nature of the Buddha that resides within all sentient beings equally, described as eternal, blissful, pure, and a true self. It is characterized as a hidden treasury obscured by mental afflictions; once those afflictions are removed, buddha-nature shines forth unimpededly.
The most influential Chinese translation was made by the Magadhan Indian monk Dharmakṣema (c. 385-433 CE), completed between 421 and 430 in the Northern Liang kingdom. His "northern" edition runs to 40 fascicles, roughly four times the length of the earlier six fascicle version, and became the most important and popular edition in China, Japan, and Korea because it promoted the universality of buddha-nature and the capacity of all beings to attain Buddhahood.
Icchantikas were a class of beings who rejected or were hostile to the buddha-nature teaching of the Nirvana Sutra, derived from the verb icchati meaning to claim or maintain, making them essentially dogmatic or opinionated persons. The sutra is ambivalent on their fate: some passages compare them to scorched seeds incapable of sprouting, while the longer Dharmakṣema translation indicates that if an icchantika gains faith in buddha-nature, he can escape the bad destinies and is no longer called an icchantika.
The Nirvana Sutra was the basis of the Chinese Nirvana School during the Southern Dynasties (420-589) and the Liang dynasty (502-557), and its teachings were absorbed by the Tiantai and Huayan schools. In Japan, it influenced Zen, Nichiren, and Shin Buddhism; Dōgen's view of buddha-nature is grounded in it, and it is one of the most important sources for Shinran's Kyogyoshinsho.