1. Movie. If I had to pick one, I might opt for Old Boy. The deeper point is that you should watch them all. If it is a Korean movie, and you can get your hands on it (as a non-Korean), it is probably excellent. This is a remarkable regularity and a good selection filter for exploring. This list is one place to start but not to stop. The Host is a fun spoof of monster movies and Shiri is a gripping thriller. Watch any Korean movie which Scott Sumner recommends. Even some of the clunkier Korean movies, such as The Housemaid, are better than most Hollywood fare.
3. Actor: John Cho, of Harold and Kumar fame. It is unlikely he is the best.
4. Video artist: Nam June Paik.
7. Poet: Ko Un is the only one I have read. It is hard to judge any poet outside of his or her native language, but I definitely had the feeling something was there.
8. Novelist: Kyung-sook Shin is alas the only one I can name.
9. Movie, set in: Korean movies aside, you might consider Pork Chop Hill.
10. Painter: Contemporary art is a rich vein for South Korea. This catalog is one good place to start.
The bottom line: This is an impressive showing and it is improving rapidly. The killer categories is movies.