I have to be honest and say the name doesn't make a lot of sense unless she's an out-of-towner originally(well, going by that backstory adopted by them), but kitsune are all Japanese.

As for backstory, her having been a feral child doesn't really makes sense if she was just in the middle of the Imaginary District(and admittedly I'm not sure how well "feral child" and "kitsune" mesh).

That being said it's easy enough for her to have been an orphaned child or something who got taken in by a new family.

I'll also say though that I think too many people are going for swords. ^^; Maybe give her a naginata or something?