Why does she get all sensitive?

I do think she's being melodramatic. But it also depends on what you say. I could give a better answer if I could actually observe one of these playful arguments. Joke, but know where to draw the line and don't say too mean things, even if she says horribly mean things to you. Girls are different, they can give but not take the meanness. I know it's weird and unfair, but if you like her then you have to be aware of this. If she gets too mean then just act sad and playfully tell her she hurt your feelings.