Short answers, pulled from the story.
Buddhism's arrival pushed Taoism to organize into a more structured religious tradition and sparked centuries of exchange between the two faiths. Deities crossed over between the two systems, including the Bodhisattva Marici, who is often merged with the Taoist goddess Doumu. Chinese Esoteric Buddhism also adopted Taoist elements directly, drawing on the Lo Shu Square and the I Ching in the design of the Mandala of the Two Realms.
Confucianists viewed Buddhism as a nihilistic worldview that damaged society at large. Neo-Confucian philosophers Ch'eng Hao, Ch'eng I, and Chu Hsi attacked Buddhism on historical, cosmological, metaphysical, and ethical grounds. Chu Hsi, known as the great synthesizer of early Neo-Confucianism, built on the Ch'eng brothers' foundation to help Confucianism overpower a once-dominant Buddhist system.
Prince Shotoku made Buddhism Japan's national religion, and it merged with the native religion Shinto over the centuries that followed. Shinto gods were treated much like Hindu gods within Buddhism, and the sun goddess Amaterasu was equated with an earlier incarnation of the Buddha Vairochana. Shinto and Buddhism were officially separated during the Meiji Restoration, though most postwar Japanese families still identify with both.
Korean Buddhist temples were often built on or near shrines to the mountain spirit Sansin, a figure from native shamanism. Most temples include a small shrine room called a Sansin-gak, typically set at a higher elevation than other shrine rooms. Offerings there include candles, incense, water, fruit, and Korea's rice wine, makgeolli.
Hindu schools such as Nyaya, Mimamsa, and especially Vedanta criticized the Buddhist concepts of sunyata and anatta. The Advaita Vedanta scholar Vidyaranya laid out these irreconcilable differences in his work the Panchadashi, defending the Vedanta doctrine of atman. Vidyaranya argued that doubting one's own existence is incoherent, since a doubt requires an existing doubter.
Buddhism originated in the Indian subcontinent in the 5th century BCE. It spread to East Asia by the 2nd century CE, later intertwining with Taoism, Confucianism, Shinto, Korean shamanism, and Hinduism.