When and who chose to move the Family to Vegas and away from New York?

If Moe Green, Klingmann, the boys from Cleveland were already there, and backed by Roth, isn't that stepping from frying pan to fire?

Michael decided to move the family to Nevada, probably with Vito's blessing. The move was crucial to Michael's long-term goal--obsession--to become "legitimate" by leveraging the Corleones' dominance of gambling and channeling it into legalized gambling. At that time (mid to late '50's), gambling was legalized in only two places in the Western Hemisphere: Nevada and Cuba.
Michael simply decided that he was going to be Numero Uno, and that nothing--not Moe Green, not Klingman, not the Lakeville Road Boys, not Hyman Roth--was going to stop him. He swatted Moe like a fly, and Roth accepted it. He could easily have swatted Klingman, the Lakeville Road Boys, and anyone else in Nevada, too. But he chose to curry favor with Roth, because Roth controlled gaming in the other place Michael wanted for his empire: Havana. So, in the deal he and Roth ostensibly agreed to, he'd become Roth's heir-apparent in Havana, and agree with Roth's plan to "give a piece" to the Lakeville Road Boys, the Penino brothers, etc. If the deal had gone off as planned, Michael would have moved them out later. But for the present, he needed Roth's assent because Roth was tight with Cuban President Batista.
The great irony was that Michael and Roth were playing each other like violins: Roth playing Michael's lust to become "legitimate" by becoming Mr. Big in Havana as well as Nevada; Michael playing Roth's greed for the $2 million to buy time to find out who the traitor was in his family.
That's one of the reasons why GFII is such a great, great film.

As an aside; Vito died before they could move to Nevada. I would guess that Vito would not have left New York. For that matter would Michael have left? Michael makes a comment in GF that he wants Carlo to be his right hand man when they make the move to Nevada. We can assume now that the comment was made strictly to put Carlo at ease while Michael confirms his suspicions about Carlo. But that statement could imply that Michael was moving to Nevada and not staying in NY, but it could have just meanet that "we" was the business and not necessarily Michael in the flesh. Then Pop dies and plans change.

That's a good question, MC! The movie leaves it open, suggesting (at least to me) that Vito's death was the decisive factor.
But the novel is clearer: Even before Vito dies, he and Michael have a chat in which he says that he's going to make his moves after Kay has the baby and to give Hagen time to be settled in Vegas. He estimates that it'll be a year from the time of that conversation. Vito appears to have doubts: " 'You've prepared for everything?' He did not look at Michael when he said this." Michael seems to sense the Don's diffidence: "You have no part. You're not responsible...I would refuse to even let you veto. If you tried to do that I would leave the Family and go my own way."

Kudos TB. I don't have the recall of the book that you do. It does answer many of those nagging little questions and ponderances (is that a word?). I can't see Vito ever leaving his beloved NY. Could he possibly get a good cannoli in Nevada?

Thats why Clemenza stayed behind. lol

Thats why Michael moved back to New York. Also what exacerbated his diabetes.