Competitions of mixed martial arts have been known to have existed since the Greco-Roman era. Modern MMA was begun in the 1920's with Brazil's Vale Tudo (Anything Goes) sport.
The tournament rules are determined by the people running the competition. They will set the age limits and categories for the participants. To go to tournaments you could be any age.
Certainly not! Learning a martial art is not related to age, it is related to the mind. Some of the most skillful practitioners I know did not start until in their 30's and 40's.
I don't know... it depends on how limber you are. You could take the basic karate. If you think you are strong enough, you could take tae kwon do. If you really need to defend yourself, do ju jistu...