I’m writing something for the class, it’s a matter of interests and all
Answer 1
They are names, not proper names. Your teacher doesn’t know what he’s talking about. Animating an abbreviated Japanese pronunciation of the word “animation”, and I don’t see why that fact would mean it should be capitalized.
However, that’s a moot point – if you’re in that class and the teacher insists it’s capitalized, then while you’re in that class, you have to capitalize it.
Logically, your teacher should also capitalize “role-playing” when used in a sentence.
answer 2
Your teacher is wrong. Anime is not capitalized (unless it’s at the beginning of a sentence) because it’s not a proper noun.