Is it common to have multiple instances of tendonitis, in different areas, as a teen?

Debatable. It would depend upon that teen's activities, levels of activity, training, supervision, and overall health. Disuse or miss use can cause this multiple site problem. Lack of stretching, warming up, and impropper use of technique can lead to this. See a sports medicine doctor and a certified trainer. Seek proper treatment of current ailments.
Not really. In a teenager make sure there is no viral syndrome, thyroid disease, lyme, inflammatory arthritis or a overuse sports related issue. Talk to the child's pediatrician soon.