I'm a bit confused by your question. I can think of... probably hundreds of example of religious intolerance and conflict in Asia across all of history. But that doesn't sound like what you're asking.

I can think of... well, a lot of examples of Asian countries sponsoring state religions (Gelukpa Buddhism in Tibet, Drukpa Kagyu Buddhism in Bhutan, Shinto and Buddhism in Japan, Sakya Buddhism in the Yuan Dynasty, Confucianism in Joseon Korea, Wang Mang's Confucian Extremist experiment...). I'm not sure if that's entirely what you're asking though, what Asian countries had state religions, because the simple answer is basically all of them did. You may have had an empire here and there with varying degrees of religious pluralism and sponsoring multiple religions (the Mongols were famous for this) and rare incidents of outright holy-war, but again, I'm not sure if this is what you're asking.

Your question is far too broad to answer in any cohesive way. Asian countries have had state religions, or form of state religion since as early as I can think of. Though their idea of "religion," what it means to be "religious," and how accepting they were of people with other religions in their borders has changed over time.

If you are more specific (geographically, temporally, or want to clarify what you mean by "state religion for all of it's citizens" then I can narrow it down).

As for that specific requirement, no? I guess there never was a state that demanding 100% total submission of it's population to a single religion. But then again, I struggle to think of any country that ever did that. If you narrow it down, or provide this extreme example of a European country that did the same, your question might be easier to answer.

Probably because Monotheism was not relevant in Asia. Popular local religions like Buddhism, Jainism, Taoism and such are not Monotheism and more tolerating against other religions. It should also be noted that majority of China "religions" are more like a way of life than true religion, like Taoism and Confucianism. The upmost, highest echelon one should worship is not God but the emperor. It's the dynasty to decide which religion the people should adopt and of course, no Monotheism because the only one people should worship is the emperor himself. Due the influence and the teaching of Confucianism that the people should worship the emperor the hardest, even harder than your father, China, Vietnam, Korea and Japan all adopted it as the main religion together with their own version of Buddhism. And when every important nations in the area follow same religions then what is the point of waging war in the name of religion?

With that said, Qin dynasty, long before Qin Shi Huang was born, had been promoting Legalism and enforced it on people, included the nobles. Qin Shi Huang himself had purged the Confucianists on large scale.

Buddhism and Hinduism also purged Jainism very hard. And who can forget the Manichaeism, the religion which once was almost as big as Christianity and Muslim, had also been purged hard in China. And also there were several large scale rebellions of Muslim community in China.