Possibly. Many skeletal malaligments begin in the feet. They are our foundation. While it's less common for children to have pain from these types of postural issues, it's still entirely possible. Overly pronated feet often respond to custom biomechanical orthotics. See a podiatrist and have your child evaluated for them.
Maybe Orthotics. It sounds like you may have a biomechanical instability in your feet that is passing problems "up the kinetic chain." custom made prescription orthotics, made by a certified lab, under the recommendations of a podiatrist can possibly help. I haven't heard of too many patients complaining of shoulder pain related to pronation though.
Definitely. Always remember that the human body is a kinetic chain. Imbalances in one area can manifest as pathology in another.