I've always believed that Sol went to Barzini first. Barzini knew that Vito would say no if he knew that Barz was Sol's partner, so he advised him to go to Tattaglia: "Tattaglia's a pimp--Vito's not afraid of him." Barzini had nothing to lose: if Vito said yes, Vito's capital and police/political protection would help make Barz, Sol's hidden partner, rich and powerful. If Vito said no, Barz knew that Sol, as a self-respecting Sicilian, would try to kill Vito. If it succeeded, Barz would be on top. If not, the Corleones would take their revenge on the Tatts. So, Sonny's gaffe or no, Barz thought he had his bases covered.
2. Because I believe in the above strategy, I think Barz talked to Philip Tattaglia just after Sol approached him.
3. I believe that Vito sending Luca to make nice to Bruno was the ultimate signal that the plan should go forward. It was so transparent that Barz, Tatt and Sol knew Vito was running scared--and not thinking.