Men and women do compete against one another in Olympic equestrian and sailing events.
Actually it is not wrestling, it is Judo and the athlete is 481 lbs.
Women were first allowed to compete in the Olympic Games in Paris in 1900. Women were allowed to compete in tennis, as well as golf.
There is no specific sports allowed and not allowed to compete in the games. The selection is based on the current populairity of the games
Tom is an Olympic High Diver.
Karate is not an Olympic sport.
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.