If you have a beard, you are not allowed to compete in Olympic boxing. A clean shaven face allows cuts to be seen on the fighters face. Beard hair can also irritate an opponent's face. The rule was put in place to protect boxers and provide an even playing field.
figure skating
Tom is an Olympic High Diver.
Baseball
Diving events.
The only event at the very first (classical) Olympic games was the Stadion foot race. A Stadion was about 200m, and women were not allowed to compete, so it would be fair to say that the Men's 200m is the oldest ongoing Olympic event, and therefore Athletics the oldest ongoing Olympic Sport.
Karate is not an Olympic sport.
equsitrian
Yes. I m not sure if it still is, but it used to be an olympic sport.
He was an Olympic swimmer.
the only sport women can do on their own with no men is naked oil wrestling
The equestrian competitions are the only Olympic competitions in which men compete against women.
Every sport is different. Certain sports have always allowed professionals via. sponsorships. Others just require athletes to compete on world curcuits. Still, many north American team sports are either allowed or disallowed depending on the league they are a part of (ie. NHL, NBA, MLB).