Yeah, go with your gut instinct. If you bothered enough to post here, you must have some doubts in your mind. Why the sudden interest in shooting? Who knows, maybe she just wants attention. However, the prior mental instability accompanied by the drug usage and relatively young age seem to raise some serious red flags. I know all this is hard to balance with your pro-gun beliefs, but some things are obviously more important than that. Take your time and I hope you can make the right decision. Good luck.
PS-If it came down to it, I think you could actually prevent her from getting a gun or a permit due to her past history. All it would take is a call to the proper authorities. I know this seems like a low thing to do, but only family will actually go out of their way to get this involved. Family first, public opinion---well, last...