Can I use "teaching assistants" to refer to students teaching peers in lower grade levels as their extra activities? What is the best word in this case?
A tutor refers to anyone who helps students with their studies, especially in one-on-one situations, as a supplement to the tutee's education. The phrase "peer tutoring" is often used to describe a case where students are tutoring other students. However, this does not specifically refer to students younger than the tutors.
At least in the US, a teaching assistant is generally an aid to a teacher or professor for a course. They are typically also a student and they may have opportunities to teach or help the students, but they may also just be grading papers, etc. My understanding of your question is that you are looking for someone who assists the students outside of classes, and a teaching assistant would not be the appropriate choice unless they were helping with a specific class to take over some of the teacher/professor's responsibilities for the course.
EDIT: Sorry, I see now that I may have misunderstood the question. By "extra activities", I thought you meant that the they were assisting in school subjects outside of normal school time. If that's what you mean, then see my answer above. If you are talking about a student in an extracurricular activity assisting the younger participants, such as in a dance class, then I refer to these as "student teachers" since they are students that are helping to teach. However, this is case, "teaching assistant" or teacher's assistant" would probably be fine as well, since they are assisting the teacher, even though it is an informal context.
"Teaching Assistant" is often a formal job title, and at the university level is used for a senior student (generally an M.A. or Ph.D. candidate) who is compensated. "Student teacher", as others have stated, is often used for future teachers' practicum at the elementary or secondary level.
I would recommend "tutor" or "peer tutor" for an arrangement where the mentor is neither compensated nor working towards a permanent position.