He was supposed to step aside for his brothers I think, according to paterfamilias.
We also won't see a Cuban Missile Crisis, Nixon was a well known anti-Communist whereas Kennedy was seen as more as an unknown. (despite having strong anti-Communist sympathies himself). Khrushchev won't put pressure on Nixon because Nixon is a more identifiable quality. As for a Bay of Pigs war...bye, bye Cuba. Khrushchev would be force to throw Castro under the bus as Nixon won't play around.
I think you're underestimating the incredible obstinance of men like Nikita Khrushchev. You are looking less at "bye bye Cuba" and more at "bye bye world".
I think if JFK lost in 1960, he'd be done. He might stay as a Senator for another term or so, but like Historyman said, the best chance for the Kennedys is either Bobby or Teddy. JFK might live to see the early- to mid-70s, but I'm not sure about after that. Nixon would likely have won the 1964 election through inertia, so 1968 would be the best chance for the Kennedy family.
I could see Bobby running, but it also depends who the GOP challenger is. Nixon would have immense pull on who the GOP candidate would be. I could see George Romney, William Scranton, probably Reagan as the Goldwaterite-type candidate, all in the race. Volpe and Lindsey could have thrown their hats in the ring, maybe Charles Percy as well. But I think after 1953-69 being run by the Republicans, the Democrats would win in '68 regardless.
If there is World War 3, then it would be more alike to 'America and Canada getting a black eye, Western Europe's teeth are kick in, and the USSR and Warsaw Pact is a nuclear wasteland.'
I think you're far too optimistic about the outcome of a hypothetical 60's WW3 on NATO's side. Yes, it won't be the total destruction that the East would experience, but it would still be the most brutal conflict in the history of mankind, and American deaths are going to number in the millions easily. A Soviet first strike (which is admittedly only one outcome, but I'd say a pretty plausible one if the US is as gung-ho in Cuba as you're suggesting) would throw about 25 ICBMs toward the US mainland - which, given a predictable spread between military and civilian targets, amounts to a hell of a lot more than "a black eye". The USSR is still going to be annihilated, the USA is probably still going to remain a major power, but at what cost? If even a fifth of those missiles detonate you've got a politically incomprehensible situation - consider the shock Japan went through after its bombing, and then triple that. You aren't dealing with a military operation any more, you're dealing with a massive sociopolitical crisis, and in my honest opinion Nixon's... "charisma" is not going to survive it.
WW3 wouldn't have been the end of the world, no, but it certainly would have been the end of the world as we know it.
Maybe I am, but the Soviets always lagged behind the United States and the West. At this point, the USSR's nuclear weapons and bombers are unreliable and inaccurate, and required a lot of prep time/fueling. I doubt all 25 even get close to the USA, or reach their target.
If the USSR dons't have any missile sites in Cuba, Moscow won't go to war over it. Khrushchev won't risk it. If there is no Bay of Pigs, and the Berlin Crisis goes different, Khrushchev has no reason to even think about putting missiles in Cuba in the first place, or can't in the first place.
I don't think the Berlin Crisis would really have gone differently either way, but in any case I don't see how that matters. There's a motive to set missile sites up, and in the Cold War a motive was tantamount to a decision. Either way, Cuba is just about the most valuable ally the USSR could possibly have. It would take a very level-headed Premier to back down from the US threatening Cuba, and Khrushchev was anything but level-headed.
OTL, it was the Russians trying to move missiles into Cuba: a straightforward first move, threatening US interests without any obvious Russian gain except MAD. ITTL, it's actual American aggression, with strong motives on both sides and two incredibly stubborn leaders who despise each other. This would emphatically not well.
The Russian interests in Cuba aside, Khrushchev push for the missile sites for two main reasons: The weakness and unknown in saw in Kennedy and the OTL screw up of the Bay of Pigs, that he can push JFK around. This is so not the case with Nixon. This is someone who had came away from the Kitchen Debate impressed by Tricky Dick strong-willed and tough-minded nature.
No, the case is a direct invasion (not even a "military advisory") in a recently-established firm ally of the USSR, which is going to last a good while - a few weeks at least, unless you really discount the ability of the Cubans who annihilated the OTL invasion - and which won't even have popular support at home. It would actually be a quite reasonable move for Khrushchev to throw some weight around, in case the American people have a why-die-for-Danzig mentality (which they absolutely will). Two steps forward, a mile back.
I really think you are overestimating Nixon's raw charisma here, and underestimating Khrushchev's stubbornness and wit.
Anyway, if there's no Bay of Pigs then there's less sign of an American willingness to retake Cuba, and therefore less expected resistance to missiles. If there is a Bay of Pigs but it fails even more dramatically than OTL, then the US will have been humiliated at a crucial time and the US public won't rush to defend any more action. If there's a Bay of Pigs and it succeeds, well, there's no Soviet ally there anyway.
I do think Kennedy did do a great job handling the OTL Cuban Missile Crisis, not bombing, or invading the island. But I do think with Nixon, we can avoid the Missile Crisis. No Bay of Pies, Castro won't lose trust in the USA and won't fear the idea of invasion. This would be enough from stop Moscow from placing the missiles in the first place. Cuba only really got close the USSR AFTER the Bay of Pigs screw up.
The PR version of the missile crisis, yes, he handled it well. And he deserves credit for not allowing WW3 to start. But it was a complete blunder on his part and it only looks good when compared to Nuclear Holocasut. A low bar indeed.