If I study for senior year HS in America, then university,will I still be considered an international student?

Because you are not a US citizen or permanent resident, you will be considered an international student, no matter where you are educated.

If you finish your education in a British school, you'll need your A levels in order to get into a US uni. If you finish it in the US, you can use a US high school diploma instead of A levels.

Yes you are still an international student when you apply to US universities.

I believe you are legally considered an "international student" AFTER you complete grade 12 and enter college/university. Regardless of such, a student visa must be obtained.