Bellydancing originated when harem girls danced for the Sultan.
Bellydancing did not originate in the harem. Traditionally in the Middle East, women dance for other women and not men. They dance with their female friends, sisters, aunts, mothers, and cousins to celebrate happy family occasions such as weddings. Even today in some parts of the Middle East, the men still celebrate separately from the women. 

This is not to say that there was no dancing in the harem. Quite the opposite, the harem women danced for each other's own entertainment and exercise.  To improve her position in a large harem or to get the Sultan's attention, a harem girl's best bet was to gain the good graces of the Sultan's mother or first wife. These were the women who had the most power and who often decided which young women to introduce to the Sultan. 

Belly dancers dance mostly with their bellies and must have a lot of “jiggle” to dance well.
Belly dance embraces women of all shapes and sizes; belly size makes no difference. In fact, if any part of the body in this dance form gets the most focus, it is the hips! To dance well, however, a good bellydancer dances with her entire body and is a visual and emotional expression of the music.

Bellydancing is easy.
Just like any other dance form, bellydancing takes a lot of practice, patience, and discipline to learn. It is an art form and, to become an expert, a dancer must learn intricate Middle Eastern rhythms as well as become an expert in one or more styles of dance. A few of the more popular styles include the classical styling of Egyptian Raks Sharki, the fast-paced Turkish style, the earthy folkloric styles of the Middle East, the floor and veil emphasis of American Cabaret, and the troupe improvisation of American Tribal. Depending on the dance style, a professional sometimes accompanies her dance and the music with zills (finger cymbals), a cane, a sword, a veil, a flaming candelabra or other props. All in all, it can be very complex and difficult.

A good bellydancer can roll quarters on her stomach.
Rolling quarters has been done by bellydancers from time to time, mostly in an attempt to please tourists who have come to expect shtick. The ability to roll quarters, however, is not a common practice nor is it taught as part of most dancers' education. The only dancer we have seen rolling quarters did so only because of her years of Yoga practice, not because she was a bellydancer.

Dancers who show their stomachs are bellydancers.
Popular American culture often fuses various cultures together to create interesting imagery and misconceptions. Because of this, many people may not be able to tell the difference between Hula, Polynesian, Gypsy, and Indian dancing all of which often have costumes which bare the dancer's stomach, but none of which are considered bellydancing. Hula and Indian dancers use their hands to tell a story. Indian dancers use their hands, eyes, and feet. Polynesian dancers shimmy a lot, but they do not perform many other moves that are essential to Middle-Eastern dance. The dances performed by Gypsy tribes have much more in common with the folk dances of Eastern Europe than with dances of the Middle East. The European dance that is probably most related to bellydance is flamenco, which developed from Middle Eastern dance when the Moors ruled Spain.

Belly dancers wear a jewel in their navel.
Navel jewels are not a Middle Eastern tradition but started in Hollywood during the 1930's to comply with censorship laws that considered the bellybutton too risqué for the silver screen. There has been a revival of the naval jewel with the recent popularity of pierced bellybuttons. Traditional bellydancers almost never wear jewels in their navels unless they are doing a comedy act where the jewel is supposed to be part of the joke or a specialty act that is intentionally trying to mimic the look of old Hollywood movies. 

Belly dancers are strippers or strippers are bellydancers.
When a stripper wears a bellydance costume in her routine, she is no more a bellydancer than she is a law officer when she wears a police uniform or a nurse when she wears a nurse outfit. There are occasions where strippers also know other dance forms such as bellydancing or ballet, but they are entirely different disciplines and should never be mistaken.