Sometimes. Like adults children may also develop blood clots. Thankfully they don't occur as often as in adults. Given the proper conditions children ill with prolonged immobility, if they have indwelling ivs (cvl), if they have certain genetic tendencies clots may occur. If a child has a chronic condition like heart disease, cancer, consult the md about the specific risks and preventions.
Possible, not likely. In a study done looking at all the children admitted to a pediatric hospital, there were only 7 kids a year over 14 years who had a DVT when they were admitted or who developed a DVT while they were in the hospital. The most common risk factors were having a DVT before, having an abnormal clotting tendency, having a severe medical condition, cancer or infection, or a long intensive care stay.