Is 아바타 the last airbender an anime?

No.
아바타 is my new 가장 좋아하는 Melodic Death Metal band.

No, 아니메 is made in Japan, I believe 아바타 was made in America

In a way, no. 아바타 wasn't made in Japan, so in countries other than 일본 it is not considered an anime. But in Japan, "anime" refers to any form of 애니메이션 regardless of nationality. Therefore, in Japan, it is considered an anime. It is not considered an 아니메 in other countries such as the United States.

its combined with a 아니메 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 아니메 refers to 애니메이션 made in 일본 and 아바타 was made in America it would make 더 많이 sense for it to just be a cartoon.

Yes it is ._.
Because it is animation.

아니메 means Japanese 애니메이션 so no it most certainly is not an anime.

Well, "anime" is the Japanese word for 만화 and animation, so, in a way, yes, but 아니메 really refers to the Japanese made cartoons, so in another way--no.