I know for some people it's normal to still be in love with an ex, but is it normal for me to not want her to be with anyone else if I can't have her? I can't even imagine someone else touching her or her touching them. I can't quit thinking about the day she meets someone, and it causes me to be really, really sad and down. She is okay with me seeing someone though, and I don't understand how when she claims how hard this break up is and she still loves me.

It's called jealousy, and it's fucking normal.
Though you should understand that she is not your girlfriend any longer and you should attempt to let it go. It will take time, though.
you just gotta get over this, time will help you forgetting her and your jeaulousy