Re: Should I ask her out?
Just do it. What do you have to lose? At worst, she will say no. No big deal - she won't place an ad in the school newspaper saying she turned your down.
Don't let shyness ruin what might be a nice relationship.
I do agree that it might be best to at least say hello and have some small talk before springing the question, but even asking her for a Coke between classes to discuss class notes would be a fine way to get to know each other.