Serving jury duty is a civic responsibility citizens owe their communities. This service ensures the proper running of the judicial system which keeps disputes from becoming violent and keeps a...
A small child doesn't have duties as such.
The adults raising that child have the duty of teaching the child that they are safe, loved and will always have somewhere to run when they're worried or...
If you live in the US, children have the same basic human rights as everyone else (ie they have the right not to be beaten, tortured, etc), but other than that their rights are extremely limited....