Piggyback Question: When/how did Buddhism, considered a foreign and barbarian religion by the Chinese upper class, become so entwined with Chinese culture?

Compare Buddhism to Christianity. Christianity was a minority oppressed religion until various Roman Emperors and Medieval Kings adopted it as the state religion. It became so widespread underground thanks to various missionaries who spread the religion among common people.

Buddhism was very similar. It might be a little of a stretch to refer to Ashoka as the Buddhist Constantine, but I think it communicates a point. He was the first big name to conquer large tracks of land, establish the Dharma as law of the land, and patronize the Sangha.

Other countries had similar events happen. In Tibet there was Songtsen Gampo who brought in Buddhist Princesses, teachers, and translators to his country. In Bhutan there was Sindhu Raja. I don't know as much about China, but the T'ang were the first of the Five Great Dynasties (Han, T'ang, Song, Ming, Qing) to propagate Buddhism and spread it across what we know of as China today. Prior to its acceptance in China, Buddhism was spread by missionaries along the Silk Route. Many central Asian Islamic countries we think of as hotbeds for terrorist activity today were once bastions of Buddhism (the Bengal was the origin of Tibetan Buddhism, the Palas propagated the Vajrayana school which was translated to Tibetan and kept safe there for seven centuries after the Muslims pillaged northern India). In Korea Silla adopted Buddhism as a state religion, and the Goryeo after them (Joseon tried to undo that work by practically persecuting them in favor of Neo-Confucianism). I don't know... well practically anything about Japan but I imagine that just like parts of India, Tibet, China, and Korea, Southeast Asia and Mongolia, there came one dynasty that converted to Buddhism and made it the state religion.

Mongolia came late to the game, but established favorable relations with the Tibetans in the 1200s under the Borjigins (Chingghis Khaan's family). Later generations would follow the Dalai Lamas. Mongolia today is mostly Gelukpa Vajrayana Buddhist.

Southeast Asia from Burma to the Mekong follows Theravada Buddhism. Various Burmese, Thai, and other Kings adopted the religion as missionaries were busy spreading it to the common people.

As far as I know - though hopefully someone can come along who knows more about Jainism - Jainism never achieved that status among an Indian - or any other - Dynasty, nor were Jain missionaries either as concerned with spreading the faith, or as successful as the Buddhists. Jainism is, after all, much harder to agree with. When it was originally founded, Buddhism was marketed (and I use that term lightly) as the Middle Path between the Brahmins and the Jains. The Brahmins believed enlightenment was contingent upon caste, while the Jains agreed with the Buddhists that it wasn't. The Brahmins agreed with the Buddhists by believing mind and intention was more important, while the Jains believed action was (hence why Jain ascetics eat only fruit, wear cloths over their mouth to prevent inhaling bugs, etc.). I'm simplifying the difference between the three religions, but you get the point.

Less people converted to Jainism, and more rulers converted to Buddhism to spread Buddhism across Asia.