There's no reason why not. Look at So You Think You Can Dance - all the dancers there have to learn several different genres. A ballet dancer has an ideal dancer's body so she's better placed to do other dance styles than anybody.
Actually, I think belly dancing would be good for a ballet dancer, because ballet can make your torso too stiff. Belly dancing gives you an ability to loosen up, and gets you grounded.
If you're studying ballet seriously as a career, you may not have time for belly dance classes - that's the only potential snag.
And no, to answer your other question - I have performed on stage but I've never been famous!