Translation:She is my classmate in Chinese class.
Translation choices are context dependent - we aren't given enough info here to say "She is my Chinese classmate" is wrong. If I were introducing someone who was not ethnically Chinese and said "She is my Chinese classmate", people would assume we study Chinese together. Just like if you said "She's my French classmate" or "She's my biology classmate". A native speaker would try to avoid saying class twice - "Chinese class classmate" sounds awkward.
The construction of this sentence is very bad. It should be:
"She's a classmate in my Chinese class."
"She's a student in my Chinese class."
"She's in my Chinese class."
There is no problem with saying "class" twice. If you really wanted to avoid it, you could say "course" instead, but it's not necessary.
Most US English speakers wouldn't bother saying classmate because if someone is in your class it means they are also a student or you are their teacher. It's pretty easy to sort out what role the speaker is in based on the context. "She's a student in my Chinese class" or "she's in my Chinese class" is how most would say it.
In order to make the English sentence sound natural I think it's necessary to rearrange it a bit so it's less literal. I would say something like "She is in my class for Chinese" or "She is in my Chinese class" - though obviously Duolingo doesn't accept this and never will. Just for the record though.
'She is my classmate from the Chinese class' is a little clumsy, but should be accepted; it was corrected to '(...) from my Chinese class'. I don't think the 'my' is quite necessary there, it comes out from the meaning in both English and Chinese sentences. The sentence kinda logically implies that it's my Chinese class in either case, with 'my' or without it.
What's the head mistress got to do with anything? She's probably in her office doing admin work rather than actually teaching anything!
The teacher is in the classroom, but no native speaker would interpret "she's in my class" as referring to the teacher, because if you did mean the teacher you'd say "she teaches my class".