To add to the other useful knowledge contributed, it's worth remembering that China and Japan have their own weird cultured/rotten foods, just made from soybeans instead of milk. Japan has natto, and China has stinky tofu, which... lives up to its name.
There are a couple of places where you can get cheese in traditional Chinese food-- there's a small area in Sichuan that eats goats' milk cheese. Sichuan has a lot of different ethnic groups, so even the Chinese food is going to be pretty different in the more isolated areas, where it's received a lot of other influences.
On top of all that, as dicrisci noted, cheese is coming into East Asian food. A lot of Japanese bar food and snacks are now involving cheese, including some weird ones like gyoza with cheese, or ramen with cheese. And frying things in butter has also become a thing.
Cheese hasn't made much of an inroad into China yet (once I ordered lasagna in an Italian restaurant in Shanghai, and it came with hardboiled egg instead of ricotta. Ew.), but it has in Taiwan. I've eaten hotpot where one of the things to drop in was squares of cheese. (It was really good.) And the city of Taichung, which specializes in these flaky bun-things called suncakes, now makes some with cheesecake red bean filling. Oh! Which reminds me, cheesecake is pretty popular in Japan and Taiwan, too (and the more international mainland Chinese cities).
In general the cheeses that have caught on in these places are the ones that are soft and mild, like cream cheese or really mild cheddar or American cheese. Much like the soy products that have caught on here are the mild ones like silken tofu, not the stinky tofu (it smells like an open sewer!).
Hong Kong, of course, has had cheese for ages, thanks to colonialism. I think Pacific Cafe in the ID has a rice bake that has some cheese in the sauce.
As for other dairy in China/Japan, neither country has historically done a ton of cattle cultivation. There's not really room in Japan (plus meat was periodically outlawed whenever Buddhism was extra-popular), and most parts of China went in more for pig farming. The exception is the north, where there hasn't been much milk or cheese, but drinking yogurt is really popular, and has been for a long time. Even now, several of the major dairy companies in China are owned/run by people of Mongolian descent, where the dairy areas are. But remember, Mongolia hasn't been part of China for most of its history.
India has had paneer for a very long time, at least according to wikipedia. I study East Asian history, not South Asian, but I'd guess that India's long history of raising cows is what led them to develop their own kind of cheese. I mean, you have all this milk, so you need to do something with it. But that's all speculation based on my reading the wikipedia article on paneer, so standard disclaimers apply.
Anyway, sorry for the long and rambling answer. I study Chinese history, so this sort of thing is interesting to me. Also, I like to eat!