The shout Alduin uses to resurrect dragons. Would the Dragonborn or other dragons be able to learn it or is it solely for the World Eater? Could it only be used to resurrect dead dragons since their souls stay within their corpses after they die or would it have an effect on other things as well?

As Dragons do not seem to be able to understand the concept of Dragonrend, I'd find it hard to believe that others, even other dragons, would be capable of utilizing the Shout. It seems something unique to Alduin, and his connection to Time.

There's no reason why not. Thu'um is Thu'um.

I always thought of it as Alduin-only, especially since it is not demonstrated by other dragons (to my knowledge at least).
Like Dargor said, it's likely something akin to Dragonrend in that regard.