Is アバター the last airbender an anime?

No.
アバター is my new お気に入り Melodic Death Metal band.

No, アニメ is made in Japan, I believe アバター was made in America

In a way, no. アバター wasn't made in Japan, so in countries other than 日本 it is not considered an anime. But in Japan, "anime" refers to any form of アニメーション regardless of nationality. Therefore, in Japan, it is considered an anime. It is not considered an アニメ in other countries such as the United States.

its combined with a アニメ and a cartoon so its a cartoon and a anime.

Not technically. It has an anime-like drawing style. But it wasn't made in Japan.

But kelly1412 has some really cool anime-avatar fanart.

It's sort of a confusing choice between cartoon and anime, since アニメ refers to アニメーション made in 日本 and アバター was made in America it would make もっと見る sense for it to just be a cartoon.

Yes it is ._.
Because it is animation.

アニメ means Japanese アニメーション so no it most certainly is not an anime.

Well, "anime" is the Japanese word for カートゥーン and animation, so, in a way, yes, but アニメ really refers to the Japanese made cartoons, so in another way--no.