[Kate*]

Ok, so this girl annoys you, believe me I've known people like that, and it can be frustrating. Here's the thing though, Like others have said, you don't know what's going on behind the scenes. It's really none of your business what she says or does or why, and there's probably very little that you can actually do about it. Unless you want to help her fit in better, just let her be. There will be people in the world who annoy you, the sooner you learn how to deal with them respectfully, the easier your life will be.

I can't say from this whether you're bullying her or not, but you are being rather harsh. It's fine if you don't want to be friends with her, and you did try to be nice just be gentle about telling her. If you don't want her to hug you, that's okay, just ask that she please not hug you, that it makes you uncomfortable.