Would you rather have a teen on birth control or have a teen missing school because she’s pregnant? They really will figure out ways to do it regardless, you were a teenager once 
She clearly trusts you enough to ask so take her seriously and be proactive. Teach her about condoms and STI/STD prevention too.
There’s two options realistically. You give her the birth control and she’s safe, or you don’t and she sneaks around behind your back and is risking unsafe sex.
I would first get her checked for MTHFR,and any other type of blood clotting disorders. As long as she doesn’t have any of that…yes.
Yes but I would emphasize sex disease, and how it will affect them mentally.