When I say 'She teaches high school English', do I use H and S as capital? (American English)
No you do not. At least not in general.
If what you mean is that she teaches English at high school, then neither 'high school English' nor 'high school' are themselves proper nouns and so are not capitalized. "She teaches high school English." is correct. This is going to be 99% of cases.
You might use capitals if she teaches a specific course, whose formal title is "High School English".
Under more bizarre circumstances you might capitalize if she taught English at a specific school called High School (presumably in the little-known town of 'High').
If a particular place is identified in the sentence, it would be a proper noun, and therefore capitalized.
Ex. She teaches English at ABC High School.
But absent that, "high school" is a common noun and should not be capitalized.