Yes. Not as uncommon as you think. Usually children with RLS have parents that are affected.
Definitely. Check out RLS network foundation which has a lot of information on this. Up to 4% of pediatric patients can have rls. The risk is greatly increased if one or both parents has the condition. Some medical problems and medications can also cause or worsen.
No. Syndromes are a collection of symptoms that often come together in a significant number of cases, and some panel makes a syndrome (not a disease entity) out of it. Your kid doesn't have rls. Take to your primary care pediatrician.