Besides, I have believe that the Avatar spirit knows what it's doing. Even if Aang was unable to spawn some new Airbenders, I figure that down the road, once Korra dies and her Earth-bending predecessor dies and then his/her Fire-bending predecessor dies, the Avatar spirit would simply be reborn into a brand new Airbender.
We don't know that. We have been told that the Avatar spirit is not invulnerable; in the episode "The Avatar State," Roku told Aang: "If you are killed in the Avatar State, the reincarnation cycle will be broken and the Avatar will cease to exist." So there are limits to what the Avatar spirit can do, circumstances that can destroy it. It's not a Western-style omnipotent god that can rewrite the rules of the universe for its convenience. So if a Fire Nation Avatar died and there were no more airbenders for the Avatar spirit to reincarnate into, the cycle would most likely end.
Really, that was the whole driving idea behind A:TLA: That the Fire Nation had disrupted the balance of the world by trying to wipe out the benders of other nations, and that the Avatar could only restore that balance by averting that genocide and ensuring that all Four Nations continued to exist. If there were magic reset button guaranteeing that an extinct variety of bender would spontaneously come back into existence, then that removes the urgency from Aang's mission. The whole point was that the cycle itself was in danger, that the whole future of the world was at stake if Ozai wasn't stopped and the balance restored.
Being the Avatar ties you to all of the elements. Korra is the Avatar; I imagine her offspring could potentially be Benders of any kind, even if she was originally from the Water tribe.
I don't think it's supposed to work that way. Avatars have access to all the elements, but each one is part of his or her nation, which is part of the point of the cycle. If every Avatar were "element-neutral," that would kind of eliminate the need for the cycle where each successive Avatar is part of a different nation, each one bringing their people's own distinct elemental character to the cycle and being different from the Avatars that came before.
Essentially an Avatar is a blend of two spirits, the innate spirit of the individual, including their nation's distinct elemental character or volksgeist, and the Avatar spirit which leaps from nation to nation in a set cycle. The former gives every Avatar a strong affinity to their own nation, while the latter gives them the ability to move beyond that. And as I've said, there is evidence of a more physical, hereditary component to bending ability as well. So when it comes to procreation, I'd assume they'd follow the same rules as everyone else and could only produce heirs of their own inborn nation/bending affinity.
How many Avatars do we know of that actually had kids?
http://avatar.wikia.com/wiki/Avatar#Trivia
The only Avatars mentioned to have descendants were Aang, Kyoshi, and Roku. Kyoshi had a daughter, Koko, with an unknown husband; Roku had children with Ta Min, of whom Ursa is a descendant (thus making Zuko and Azula descendants as well); and Aang had children with Katara.
Still, those are the past three consecutive Avatars, so it would be rather a large coincidence if they were the only ones. It suggests that it's fairly routine for Avatars to marry and have children.
