Why would Deathwing fight the Sha?
They're both Old God aligned.
The Sha: As fragments of Y'Shaarj (or echoes as the Klaxxi put it) they are definitely aligned with the Old Gods.
The Old Gods and Klaxxi
Though the Elemental Lords part is wrong; they fight with each other because that's what they do.
When working towards a specific goal the Old Gods put the games on hold and work together.