Which age for braces will have less risk of root resorption? Young or old? Does age have anything to do with root resorption? Is that true the younger you are, the shorter the teeth root is and so the more risky for root resorption with teeth movement?

Always. There is always root resorption associated with tooth movement. Resorption is almost always microscopic and not of concern. Noticeable resorption can be caused by heavy intermittent force, movement of roots into outer layer of bone (cortical plate), genetic predisposition, and many other reasons. Timing of Orthodontic Rx has many determinants. Please see a qualified Orthodontic Specialist.
Root formation. The completed formation of the roots is the determiner, not the age of the patient. Roots are usually completely formed about 3 years after the tooth erupts, so there is a slight variance between patients. Your concerns are valid and should be discussed one on one with your child's orthodontist. They will have the radiographs and the experience to treat your child appropriately.
Regardless of age. The etiology of root resorption is multifactorial and consist of individual biologic characteristics, genetic predisposition and the effect of orthodontic forces. Ask your orthodontist for more information.