First of all, Korra is not black. She's brown. There are plenty of brown asians. She is the same color as Sakka, Katara, Tarlock and every other water bender. If you look at eskimos, southeast asians and filipinos, a lot of them have the same skin tone as Korra. I'm Thai and my tone is closer to Korra and Katara's and I have NEVER been mistaken for a black person. Mexican, maybe, but that's because I live in San Diego.
Secondly, IF she was black (which she is not), so what? There should be no problem with that either way. Maybe there should be some black characters written into the series. That would be badass.
I thought it was really cool that they had brown characters, because that's something we seldom see in a series - the in-between colors.
Finally, OF COURSE AVATAR KORRA IS A GIRL! Look at the cycle. Aside from water, earth, fire and air, the pattern has been male, female, male, female, and so on and so forth.
EX: Avatar Roku - Fire, Male
Avatar Kyoshi - Earth, Female
Avatar Kuruk - Water, Male
Avatar Yangchen - Air, Female
The next Avatar after Korra (if there is one) will be an earth bender and a male, according to the pattern.
Sorry I got so heated, this topic just boiled my blood a little bit.