Yes, some members of the Japanese imperial family did marry into other royal/imperial families.

Princess Nashimoto Masako (it's wikipedia, but this is a pretty basic, factual question about whether she existed or not) is probably the best example.

Her marriage to Crown Prince Eumin, the heir to the Korean throne, can be taken as an attempt to enhance Japanese control over the Korean peninsula by symbolically demonstrating the "union" of Korea and Japan. They were sent on a tour of Europe in 1927 to demonstrate that unity to the world, as well.

On the other hand, there was a lot of popular resistance to the wedding on the Japanese end, including an attempt to blow up the royal carriage that carried them to and from the wedding ceremony. I don't know anything about the Korean reaction to their marriage.

Why not Russia or Britain? Better to ask why Russia or Britain. To venture a broad explanation, Russia was in direct competition with Japan in Korea, and Japan beat them soundly in the Russo-Japanese war. When Russia still had a royal family, the Japanese were more or less getting their way. There was an Anglo Japanese alliance, but for Japan's Imperial strategy at the time, smoothing the process of political and economic penetration of the Asian mainland was more of a priority than cementing relations with distant European allies. Remember, both the Sino Japanese and Russo-Japanese wars were fought over influence and concession rights in Korea.