Yes, they have residence requirements to maintain in order to keep the green card, although a child who lost his/her green card due to spending too much time outside the US can apply to have it restored if they apply by a certain age (I think 18).
Children under 18 don't follow their own naturalization process. While under 18, they can only obtain citizenship through a parent. The parent's naturalization would confer citizenship to the child, provided the other conditions of the Child Citizenship Act have been met (child is living in the US in the custody of the naturalized parent, etc.).