An increase in Japan's demand for US goods would cause the value of dollar to appreciate? An increase in Japan's demand for US goods would cause the value of dollar to appreciate? Why is that? The official answer is "Japan will buy more U.S. dollars." However, my thought was that isn't it usually the case that US companies get paid of Japanese yen rather than U.S. dollars? And hence Japan will not need to buy more U.S. goods. What is the problem in my thought?
You're just intermediating. The US companies would still have to pay their workers in USD, therefor, instead of the Japanese companies buying dollars in your scenario, the exporters are using the importers' Yen to buy dollars.

It's the same net effect.