Is 13 years old too young to take birth control if the girl is only using it to shorten her periods?

No. She is not too young...But....They are synthetic hormones and if she is taking them only to shorten her periods I do not recommend it. You will get superior results for painful and or long periods with chinese medicine and acupuncture.
Not at all. In the hands of an experienced adolescent specialist or OBGYN, the right product for her needs can be found and used as needed. However, it may not be necessary. Many of these girls are just not releasing eggs for the 1st several months and that can result in heavy or prolonged cycles. It may self correct within 6 months. Get a consult and decide what works for you.