You're aware that public schools in NC|SC are often considered to be inadequate? IMO, unless you got lucky with a particular school, scholastics would suffer.
A friend (editor in chief of an academic press) sent his daughter from Switzerland to Mississippi for a year. Not known for the best public education system. She had a great time and made some long-term friends. Given the intellectual resources in her family, falling slightly behind in school was not an issue for her in the long run, but the experience broadened her a lot and of course solidified her already strong English. Or Mississippian.
No question, most US high schools will be sub-par compared to a Swiss (and presumably Danish, though I don't know), high school. My Swiss friends with just a high-school education often feel more like Americans who have been to college.
That cuts both ways though. It means that she's likely going to be ahead of her peers in the US scholastically and will therefore be able to handle the coursework in a foreign language and spend time doing "cultural anthropology."
Then, of course, on the other side, as RC suggests, she might fall a bit behind in some straight academic ways compared to her peers in Denmark. But honestly, a smart 19-year-old could probably do high school in two years. Even at 14, I came from a crappy school to a good school and got thrown into Algebra II having had less than half of the Algebra I course and it basically went fine (I hated the teacher, but so did everyone else, including other teachers at the school, as I learned much later).
Given the household/family Raz's daughter comes from, I'm going to guess that she will have the support to make up any lost ground she incurs by traveling to an de-developing nation for high school.
