Should I Email A Random Girl? (I'm Running Out of Time)?

A better idea would be to talk to her in real life. Emailing would just be creepy... and she could just ignore you and think you're a stalker. Since she randomly greets you at random times, just make conversation with her! Haha, it's nice to know that you like your "good looks" and your smartness. Just be yourself, and if she doesn't like you for who you are, screw her. (and not in that way.xD)