Aww!! Congratulations on her! She reminds me so much of my late female, Kabuki. (but then again, every female betta I see lately reminds me of her... maybe its some subliminal sign that I need to get a female sorority tank soon.)
Anyway she is gorgeous! Though no names come to mind.
And as for her fins, basically what was stated before. My SD male had his tail chewed up pretty badly, and its healing well with me keeping his water sparkly clear and adding a very tiny pinch of aquarium salt in there.