Based on Sollozzo's planning of everything else I think he was in contact with Paulie much before Vito's shooting. Sollozzo was two steps ahead of everyone until he ran into Michael. If the Corleones can wait years to activate a plan, surely Sollozzo could have planned ahead a few months.
I believe Sollozzo was sort of an independent contractor for Tattaglia and Barzini. He would have had to clear his actions but I don't see him asking so much as almost telling them (at least Tattaglia) what he was going to do.
Sollozzo was known as a top drug guy, which would seem to indicate he was not just a soldier. I don't think a soldier could have requested and received a meeting with the entire Corleone hierarchy.
Per book, Clemenza believed Paulie was just greedy and ambitious. Clemenza thought perhaps the treachery could have been forestalled if the Family had paid Paulie more money. Or not. Clemenza didn't spend a lot of time thinking about it.