It's not required that you show your belly in class (though it is called "belly dance" for a reason). Some dancers find it helpful to wear cropped tops or sports bras to class so they can see if they are doing the movements correctly. Especially when practicing belly rolls, it may be helpful to see your belly.
NO! Anyone can bellydance.
Belly dance is probably the oldest dance on the planet. In many cultures it was a dance performed for women, by women. This dance did not originate as dance to gain the attentions of a man.
Traditionally belly dance was danced by women to entertain themselves. Remember this was long before the age of television and radio! Girls would learn to dance by watching their aunties, grandmothers and older sisters.
Some people believe women danced to show other women how to give birth - a very possible theory! Women came together to dance in celebration of harvests, marriages, coming-of-age or childbirth. And yes, sometimes men joined in and danced, too!