Tattaglia was killed as further payback for Sonny's death (despite the fact that Tattaglia's son was already killed).
Wasn't Barzini responsible for Sonny's death? It's what Carlo stated before he was whacked, that he was confronted by Barzini. Plus the Don said it to, "... it was Barzini all along" after the commission meeting.
Something I don't understand is why would Michael make such a bold move in eliminating the heads of the five families, to move to Nevada? What's the point in that? He had the city, and the cosa nostra, in the palm of his hands. I could understand expanding your influence to Nevada, but moving it entirely?