Welcome, Immobiliare! You're asking good questions!
One thing that often applies in the Trilogy is directorial license. The dramatic impact of that scene--Michael arriving at a nearly deserted hospital with no guards around Vito--wouldn't have happened if Tessio had called Sonny with the bad news about his men being arrested.
I agree that Tessio's treachery was directed against Michael, not Vito. But it was being hatched before Vito died. In the fishtank scene, Tessio starts the bitching. Then, when Vito says to Tess and Clemenza, "do you trust my judgment?", Clemenza replies, fervently, "Yes, Godfather." Tessio just hisses, "Yessssss" through clenched teeth. I think he also gives Michael the left-handed handshake.
