What about "take a seat"
You need to guess from the context which we don't have here. So it could be both.
Why is this necessarily plural? Could "take your seat" also be correct?
Because in the context of school, it is likely a teacher saying this to their class.
"Take your seat" is also correct.
I wrote “Be seated!”
Have a seat?
Sit down should be correct
"Sit on your seat!" should also be accepted I feel like
Would this be acceptable in a high school?
I put, "Sit in a seat."
This is quite informal. 자리에 앉으세요 would be much better.
I guess it depends if you speak formally or informally to your students. If they are much younger than you it's probably fine