No, the Olympics do not have a category for Karate. There is a category for Tae Kwon Do and one for Judo.
#1 learn Tae Kwon Do
Yes of coores it does
It already is/was in the Olympics
It appeared in 1992.
No, taekwondo was in the 1988 Olympics, 1992 Olympics, 2000 Olympics, 2004 Olympics, 2008 Olympics and it is scheduled to appear in the 2012 Olympics and the 2016 Olympics.
16 years old.
You can be as young as 13 to participate in the Olympics but you must register a year before because the Olympics are so important and they take a LONG time to get ready for. If you don't believe me then research Sarah Willson a 13 year old Olympic Gymnastic! To answer your question, however, then yes you can be 15 to join the Olympics with the right qualifications.
Taekwondo is a confirmed sport for both the London 2012 Olympics and the Rio 2016 Olympics.