Should children study Martial Arts?
In general, yes. Some of the possible positives would be control of aggressiveness, instilling self-respect and self-control, as well as
self-defense.
The style that a child should take is a totally different question, and is
directly influenced by the style, if any, of the parents. It will of
course be convenient if the child can practice with, or at least in the
same school as, the parents. The major issue with children in the martial
arts is the integrity and trustworthiness of the teacher and the school.
The joints and connective tissues of children are more vulnerable to injury
than those of adults. Keep this in mind when selecting a style and school
for a child, and discuss it with the instructor. Schools which allow aggressive
joint locks to be applied to children or don't train them to
refrain from snapping/hyper-extending elbows on strikes and knees on kicks
should be avoided. (It is for this same reason that good baseball coaches
will not allow young pitchers to throw pitches which require hard snapping
of the arm - like curve balls). Throws, however, are quite different; the
small size of children makes them naturals for arts which require falling
down.