We have to go with what is known, and that gets pretty murky:

What's known is that, after Luciano went to prison for double-digits in 1936, he appointed Genovese as acting Don. But Genovese was forced to flee to Italy a year later, one step ahead of a murder rap. Frank Costello became acting Don, and cemented his role as "the Prime Minister of the Underworld." Genovese wanted to resume his role as Don when he was deported from Italy to the US, but Costello was too powerful.

Apparently Luciano's presence in Cuba following his deportation from the US was ratted out. Genovese generally gets the rap, but there's no doubt that Luciano wanted to resume his place as Numero Uno in NYC. So, any of the Dons at that Havana meeting had a built-in incentive to rat out Luciano's presence.

Most historians agree that the Apalachin disaster was the work of alert cops who were suspicious of the large number of cars in Joe Barbara's parking lot. Some have said that the cops were sore at Barbara for not paying them off. Nobody has said Genovese set it up--he was one of the victims. He was definitely set up for the drug bust in 1959.

I'm not sure what you mean by Genovese's FBI connection. And, there's no reason to believe that the Valachi thing happened for any other reason than a case of mistaken identity.