What are the best sports or martial arts to introduce to a child in order to increase its flexibility and athleticism from an early age?
If you want to increase a child's flexibility and athleticism from an early age, the best answer will not be what you expect. I have watched young kids martial arts classes over the years. They basically are controlled daycare. Kids with short attention spans in groups, with their parents watching, the instructors spend a lot of time just keeping them focused.
If you want flexibility and athleticism, ballet, jazz dance, or hip hop dance are the best bets. Then, once they hit their teens, you can put them in martial arts. It will be a bit of re-learning curve for them, as the goal and intentions of their movements will be different, but if the art is right for them and they for the art, they'll catch on.
Dancing With the Stars TV show has had one or two mixed martial artists as dancers one or two seasons, and the dancers had no problem keeping up with them, and the dancers had better body movement and flexibility in addition to matching or surpassing the mixed martial artists in terms of physical conditioning.
Side note, the dance classes can have a positive effect on early social skills, but this is dependent upon the instructor.