Julian, I hope this advice serves you well, since I registered specifically in order to post in this thread. I have been moved by the situation you find yourself in, I think it must be really difficult for you, being so young and inexperienced. I say this not as a thinly-veiled jibe, but merely out of compassion, since I still remember what that was like. In fact, I don't have nearly so much experience as many people my age do, but I still believe I have something relevant to offer here, by baring my own soul.
I did not go on a date of any kind until I was 23 years old, and then I had to be set up. I was socially awkward, lacking in both style and confidence. I was also a little overweight, although that has largely remained the same. The difference is that now I don't let it bother me so much - after all, I am not a whale, just a little heftier than would be ideal. I certainly had my share of crushes through high school, but I was so inept in such matters that none of them ever got anywhere. I was also spectacularly bad at selecting my targets. Looking back with the benefit of hindsight I can recognise several girls who would have welcomed my advances, had I only been wise enough to notice.
Even when in my mid twenties I had first started to develop some confidence in order to start some timid experiments in dating, I was very insecure and clingy. To this day I remain so, to a lesser extent. It is a scar on my psyche which will heal, but never disappear, due to my early experiences. I basically managed to ruin most of the early relationships I had because of this. For this reason, I have sympathy with your female acquaintance and wish to strongly go against the grain of the advice you've received thus far.
She is not "batshit insane", and the odds are she's not dangerous either. She's insecure, and scared by feelings she's never had before. That is an emotional problem, but not one which can't be resolved. There is only one way in which she will improve her emotional coping skills, and it is the same way I eventually did - by finding someone patient enough to understand and help her. I appreciate that it isn't your duty to help her just out of the kindness of your heart, but I think you want to.
You claim now that you 'never felt the same way', but I believe that is just a self-defence mechanism kicking in, and that at first you felt the beginning of something potentially special, just as she did. I too would have been significantly distressed by the knowledge that a potential girlfriend was extremely religious, but I do wonder if maybe it is something you can work around.
You should talk to her. Meet her somewhere quiet where the two of you can talk privately, and express your concerns. You should ask her if it would be OK for the two of you to never, ever discuss matters of religion and politics. If that would be OK with her, then there's no reason it should be an immediate problem. The ideal is that you'd find someone who you could discuss these things with, but how many things in life are ideal? You need a real commitment to it though, no off-hand comments, or provocative statements.
You should also tell her that you expect that if she meets any of your friends who have opposing views, or who are homosexual, she will treat them with respect. That's non-negotiable, and she should be able to expect the same from them. Also non-negotiable. You also need her absolute guarantee that she will not proselytise to you, nor criticise the atheist position as immoral. In return, you will offer the guarantee that you won't criticise her choice of Christianity. These things are also non-negotiable, and in theory should fall under the above agreements, but it is important enough to merit a specific agreement.
It should be made clear that repeated violations of these agreements would require you to break off all contact, even friendship. The occasional breach of boundaries is understandable, but should be met with a simple response such as, "Didn't we agree not to discuss things like this?" delivered in a neutral tone. Any attempt to continue the discussion beyond that point is fair grounds for the other party to be unhappy, and to express that.
It can work, I know couples who have made it work for many years.
Regarding her multitude of text messages and calls, I don't honestly see it as a big problem. She's probably insecure, and we know she's inexperienced. She obviously really likes you, and she already knows there's a chance her religious and political views might drive you away. She has good reason to be a little worried. Admittedly behaving in that way is usually a good way to drive someone away from you, but it doesn't have to be. You could make the choice to not let yourself be driven away by any of this.
It seems to me as though you really did like her at first, and there's no reason why you still shouldn't like her. If she's willing to set aside your differences, and you are as well, then it can work. If you treat her with understanding, love and kindness, she will repay you with trust, devotion and love in return. I understand that the differences between you are a little off-putting, but you did like her before you found out these things. If you had never used the word "God" in that way, you'd still be none the wiser.
I don't see what you have to lose by dating her. She already seems to think you're in a relationship, and you say she already knows a lot about you. She probably isn't a dangerous psycho, and if you reassure her a little, she will become a lot more secure in the relationship and in herself after a while. If eventually you decide to break up, well... people do. She will probably understand, and even if she doesn't... she would probably see it as a breakup if you stopped talking to her right now.
Your best course of action is just to try it and see how things go, in my opinion. The phone sex and cybersex sessions you've shared prove that she's really into you, and that you're pretty into her too. They prove she already trusts you, and that she probably doesn't have a traditional Christian attitude towards sex. Don't see it as her being a hypocrite, see it as an example of how she is drawn to you in a way that transcends her religious views. That might even be the first sign that she won't stay religious forever. People do change, especially in their late teens. I know I did.
I doubt you're going to take this advice, but I really hope you do. It sounds to me like there is a chance this could be something great for the both of you. Even if you don't stay together for a long time, it might make a difference to the way both of you approach future relationships.
The tl;dr version: Go for it, you never know how well it might work unless you try.