Am I mean?

No, you aren't. Just make sure you don't hurt her too much, because she can have enough of it suddenly. If it hurts her so much maybe you can stop the insult-fights. And if she starts one, take the punch and leave it be.

I wouldn't say your a mean person
What you said may have been unintentionally mean, but i wouldn't call that anything but misunderstanding unless it was with bad intent
I also wouldn't say she's "over-sensitive" just more sensitive, and maybe you should just try harder to not say anything that might hurt her. It's also probably a good idea to apologize when you offend her, not for what you say, but for unintentionally causing pain :) just my thoughts on the subject

I think you should just explain to her that you're teasing her and that you mean absolutely no harm by what you're saying. She probably doesn't even realize that she says similar things to you. She sounds like someone who pretends to be confident when in reality she's not. I don't think you're mean at all, and you should relax and be yourself, just try to make it obvious when you're joking around, and also maybe be more sensitive to her at times, show her a kinder side so she knows that in reality you are her friend and you don't think anything bad about her. I don't recommend accusing her of being too sensitive to her face or anything, people have said that to me and it just makes things worse.