Hi,
Yes, there are music schools (high schools) for which you have to take an audition to get into. I know so because my mom went to a music high school. She majored in piano.
There are also job schools, sports schools, art schools, foreign-language schools, science schools etc. But they also require an audition or an entrance exam. Regular high schools usually don't require any auditions/entrance exams.
Private schools are open to foreigners, but public schools are free and only offered for Korean citizens. There are several "foreign" schools, too, which is open only to foreigners (or Koreans who lived abroad for a long time).