I'll write some more about this later.

For the moment, I'll make a few remarks: <i>"an orbifold is a topological space which looks locally like a finite quotient of Rn by a finite group of O(n)"</i> is a bad definition. For example, it fails to address the (important!) class of non-effective orbifolds. E.g. the moduli space of elliptic curves is a non-effective orbifold because every elliptic curve has a $-1$ automorphism.

Second: it is important to realize that orbifolds form a 2-category, not a category!
If you get something that looks like it's a category, you're doing something wrong.