I think you are in a really tough situation.
It seems like she came in at the beginning with an attitude of mistrust.
Everything you mentioned is really vague and if this is how she is presenting it to you, I can see why you're at a loss.
I think I would ask her specifically what she needs from you to feel included. I would ask her to give me specifics on what I was doing that made her feel unequal. Until she does a better job of communicating what it is exactly that is bothering her, you have no idea what she needs.
I think it might help to sit down with her (maybe off campus at a Starbucks) and have an honest conversation about what is going on and what both of you need.