And keep in mind that the Anti-War Movement of the 60s and early 70s was very much by the end a very organized and 'professional' group that had been built up through experiance and trial. The moved on to become the core of the Anti-Nuclear Movement and brought the same organized and professional 'protest' system to play there as well.Indeed. I also think this would have a BIG effect on the counterculture of the 1960s. Hippies rode the anti-war movement to prominence IOTL. In a TL where Vietnam was less prominent, I don't think the hippies would be as noticeable. Don't get me wrong, they'd still be there, hippy culture evolved out of the beatniks of the Fifties, but it'd be much smaller and much more confined. I think sexual mores and fashion going into the Seventies would be much more conservative. It'd probably be something along the lines of what happened in England IOTL. Change IS happening, but the establishment is definitely still in control.
It was rejected out of hand by the Soviets, politly mind you, but firmly. The Soviets would get 'nothing' out of the effort and would expose just how marginal their own space program really was and they were well aware of this. By the beginings of the Gemini program in 1965 the US was fully commited to going it alone and the Soviets had begun to fall behind which was even less incentive for cooperation.One interesting WI is Kennedy prior to his assassination was mulling over a joint mission to the moon, apparently discussing it with Khrushchev. Perhaps not very likely but would certainly make for a historic image, an astronaut and a cosmonaut shaking hands on the lunar surface.
Kennedy was never comfortable with the Lunar goal as he was well aware how much it would cost and how much of the US economy and resources would be tied up in just the first few years building the necessary infrastructure. He pitched the cooperative mission which took everyone by surprise as a last-ditch attempt to cut the US effort but when rejected by the Soviets he went all in on the US effort despite continued misgivings. By 1963 or 1964 it's probably too late to try and ramp back the US effort as the NASA budget is already pamped up and the various program items are in the pipeline so there's really no savings to be found and reduction would only make things worse.
Given a surviving Kennedy it's possible he could re-pitch the suggestion in the wake of the Cuban Missile Crisis but it would be an issue as that's what pushed Krushchev out and I'm not sure Brezhnev would have been willing to take the chance.