Mochi: Japanese. The steamed rice is pounded into paste and molded into the desired shape. Also Koreans do something similar.
Rice noodles: Uncooked rice is ground into flour and then rebuilt into a new shape.
Rice paper: Vietnamese thing. Same idea, different shape.
Puffed rice: Popular in many countries.
Rice crackers: East Asian, cooked rice flour
Arròs negre: Spanish thing, squid ink + rice
Nian gao: Chinese new year cakes
Arancini: Italian rice balls
Appam: South Indian pancakes
Jiuniang: Chinese fermented sweet soup
There are also many alcoholic beverages made of rice.

See also: OMG, there are so many

It's from India, not Japan, but biryani is amazing.

Sugar Smacks

It's Japan, so probably some wild kinky shit.

Rice paper?

And does it end up on a plate as plain rice?

Sure but we also should not exaggerate the inclusivity of "the West" The availability of international restaurants has little influence on what people outside of diaspora groups are making in their kitchen. Consumption is one thing but when it comes to household cooking a lot of the perception of rice here is limited to risotto at best and Uncle Ben's at worst. That's not even getting into the fact that a lot of the cuisines you listed were looked down upon in the West until as recently as thirty years ago (and some still are).

It’s available, but generally speaking people don’t even realize what it could be used for. Many westerners have actually eaten rice in a variety forms because of the reasons you mentioned.

When you eat at an Indian restaurant, you’ll just eat the strange little cakes without realizing that they’re actually made of rice. That’s not what people think when they hear the word rice.

What I meant to say is that rice tends to keep its form in the West. It’s cooked and mixed with a variety of different things. What you see on the plate can immediately be identified as rice.

Asian countries get very creative by turning it into flour or paste and using that like any other ingredient. Just like butter, sugar, or salt could be encountered anywhere, rice can hide just about anywhere. You would never know unless you made that food.