The criteria basically amount to two things, and two kinds of people who are considered qualified by our state to make such a change. Those being judges, and medical doctors.
The "variance" is basically just a token nod at the right to appeal. It gives you somebody who will review your case, but I can promise from personal experience that without those judge or doctor documents to satisfy the first "three" requirements (really it's two, one is gender change, two is gender+name change), they aren't going to give a damn.