How to solve this? iin?

Ok, so this girl I'm dating is really really cool, I've known her for a long long time, and these problems thing's I'm going to talk about have never mattered until now, because we decided we should be in a relationship a bit more than a month ago. Let's call her Alex. Alex is often really happy, but when she is sad she's really sad, and goes to cutting herself for relief. She takes mood stabilizers because she has a disorder called borderline personality disorder, which I don't think affects her personality at all, just how she handle's emotions because she feels the same emotions we feel, but for a longer time each time she feels those emotions, because of the disorder. It's really interesting, I would recommend looking it up on wikipedia. I really like this girl because when she's happy, it's like she's never going to be sad again, and just being around her makes me happier, which is sweet. She's a grade below me though, and she has a bad reputation because she made a few bad decisions, sending pictures of her breasts, and having sex with a guy in her grade, which created a huge, true rumor, that really hurt her because she thought she could trust that guy. It's her fault she did those things, but people in her grade don't even know who she is, and all call her names and call her a slut. Many people really like Alex, but she finds it soo hard to deal with the one's who don't because she just hears what they say over and over again, and because of that she feels worthless. It doesn't help that when she goes home her bi-polar mom yells at her, and says things like "I would rather have you commit suicide than have a bad rep", because in our town, that's all these rich preps care about. Because of these kids, her home life, and the fact that she has that disorder, those sad, trapped emotions can't be dealt with besides cutting she feels like, so after a while the school, and Alex's mom noticed, so she had to go to psychiatric rehab for about three weeks. Her cuts are nothing extreme, more scratches than cuts, and I think they are more of a way that she is depressed, because of all those reasons. At that time we were just really good friends, and it sucked to not see her for those three weeks at the skatepark, where me and her friends chill. We aren't all weird emo kids, we just don't play football, we skate, and I realized that I could get a job and put money into my bank account, instead of being bitched out by coaches. My friends, her friends, little kids from the playground, and a many other people love to hangout at the skatepark, so there are no negative influences from there, and we aren't "burnouts". When she got back everything went back to normal, and I got my license, so hanging out with her became real, besides just at the skatepark. When that happened I realized how much I like Alex, because I can say anything to Alex and know she's not going to make fun of me or judge me, because we are such good friends. Like I said earlier, she's really happy when she's happy, and since we both like each other, we both have an awesome time just watching tv, eating food, spooning, and hooking up, but not as just a hookup, it's because we like each other, so it's completely natural, and we are both chill with it. That week was the week before she got caught cutting AGAIN, and the rehab's haven't found her a bed yet, so from then until now during the day she has been going to rehab, but not staying overnight. Alex isn't allowed to go to school anymore and when she gets home, it's at 8:30ish O'clock, and we can't even facetime, just texting basically which is annoying, because random things can be misunderstood, and it's just not fun like hanging out with someone in real life is. Since I'm in a relationship with Alex, I want to see her any chance I am able to, because very soon they are going to find her a bed, and she's going to be gone at rehab for at least a month this time. I will be able to visit her a few times, but basically not know her for a month, and this brings me to why this is a problem. I will say "what are you doing tomorrow after you get home, because we could probs chill for a few hours" and she will have a reason she can't and most of the time she is telling the truth, but now I learned that yesterday, after I've been trying to chill with her for like a week, she goes to one of my friend's house, with one of her friends. I don't think they did anything bad, but I can't understand why she wouldn't tell me, and not hangout with me before him. I am always there for her, and I feel like I'm secondary to other people in her life, even though we are "dating". While we facetime she won't laugh at what I'm saying because she's in "a bad mood", or "just tired" or "on her period" then she gets a text from some random person she won't identify, and starts laughing. she's like always paused and I'm like wtf I give you 100% of my attention, not to look at "paused" the entire time. I want to know what to do with our relationship, because if I'm going to be lied to and not appreciated, then fuck that I can spend my time with a different girl. That's not what I want to do though, because I like her a lot, but I don't know why she can't tell me things like that. I'm probably going to her house tonight, so I am going to ask her about it, and hopefully all goes well, but if not, what should I do? I feel like if I leave her it would be the most epic state of depression she's been in, but still, why does she do that to me? I want to be able to trust her, because I'm never going to do anything with other girls. I have friends that are girls, and if they want to hangout, then I'm going to tell Alex, and not do anything sexual with those girls. I hope this can be resolved, but it really bothers me how she takes me for granted sometimes, and then after not seeing each other in real life for a while, she goes to someone eles's house, so idk if she is lying about the time she gets home at night even. One of my least favorite things is being lied to, because the liar think's I'm dumb, and are using me for only benefit. Is it normal to have a girl that does other things without telling you? Is it normal to have a girl that cuts herself? Is it normal to not judge her really negatively because she cuts herself? Is it understandable to be with her even though she has that disorder, which makes her really really cool but also really really sad sometimes? Are her actions based on that premises? Does she not know what to do? I should be able to answer those questions, but I can't. I don't know if any of you have answers to those questions, but if you have any experience with a girl who cuts themselves, or with borderline personality disorder, please respond.