As DC said, the novel addresses most of your first question, Capodi. Part of the answer to your second question is contained in a deleted scene from GFII. In that scene, Neri tells Michael that he's located Fabrizio, who he says came to America in 1956, "sponsored by the Barzini family." Since Barzini put Fabrizio up to bombing Michael's car in Sicily years earlier, we have to assume that someone in the Barzini family was still in power in 1956, and influential enough to make good on Barzini's years-old debt to Fabrizio. Since that action was hostile to Michael, it seems that the Barzini organization wasn't totally decimated or subordinated after Barzini was killed.