she sounds extremely manipulative, and is used to getting her way by doing this. it probably is a result or contributes to her mental state which sounds like she needs some proper help.
if she has told you if you tell anyone she will do something much worse than just hurting herself, if you tell someone, it just means that an adult or person in responsibility will be able to get to her and deal with the fall out, rather than you being stuck dating a girl because she is telling you she will hurt herself if you don't and will do something much worse if you tell people that why you're dating her.
Don't date her, don't have pity sex with her, don't do anything that makes it seem like you are giving her any sort of signals. You've told her you're not interested, and she won't take no for an answer. Don't arrange a date, don't let her engage with you about talking about arranging a date.
Tell someone, preferably a teacher or someone who knows both you and her, that she is threatening to harm herself if you don't go on a date with her, and that she is threatening to do much worse if you tell anyone about it.
I can understand you not wanting her to hurt herself or carry through her threats, but is there much chance she will actually go through with them? Years ago, I broke up with someone who was threatening to hurt himself and run away in an attempt to get back together with me. I was worried about him, but it was over between us. One day I snapped, and told him that there was no way we were getting back together and he needed to stop thinking threatening me would make it work. Funnily enough, it got through to him and it turned out all his threats he made and when he said he had left home etc. were done while he was sat playing Xbox in his room. After me, he dated another girl who I became friends with, and started going down the same very manipulative route. The poor girl was panicked and asked me what to do as she was worried, and she ended up going to his house only to find he was fine and hadn't hurt himself, and again was sat playing Xbox. He then dated my best friend at the time briefly, and whilst they were together did the exact same thing to her when they were still together, and I told her to ignore him because it was just emotional blackmail for sympathy, and to try and keep her in the relationship (we were about 14/15 so it was nothing serious at that point).
Please, tell someone. Tell them what she has said and what threats she has made, so they are able to protect both you and her. Do not put yourself through an emotionally manipulative, unhealthy relationship that will end up hurting and damaging you, just to keep a girl happy because she has said she will harm herself otherwise.