One does not automatically lead to the other. There are numerous other points of contention between the USA and USSR during this period; Berlin and Laos are the biggest, Congo to a lesser extent. Moreover, this isn't the Nixon of 1968; he'd have been elected in 1960 partly on the back of his anti-Communist record. If say a Test Ban agreement is reached between Washington and Moscow, it'd likely be a long, arduous process of negotiations.
The Oswald factor almost goes without saying, but Diem's time was running out arguably by 1960, and certainly by the time of his OTL assassination. He was simply pissing too many people off throughout South Vietnam (there's an illuminating interview in Christian G. Appy's oral history of the Vietnam War as to Diem's unintentional alienation of even the lowliest bureaucrats.) We must also remember that it is harder than it looks for the US to actually topple a foreign leader; most of the time coups of that nature are never organised by the CIA, but by army colonels or local government personnel who just happen to tell the CIA what's going on in the hope of getting a little extra support. Such as it was with Diem in 1963, and such it was with Mobutu in 1961.
Partly agree with this; the Democratic Party is going to face a serious crisis in the dealings between its conservative southern wing (who conveniently monopolised many of the Senate committees) and its more liberally-inclined northern half. A complete split however underestimates the capacity of the party to adapt, especially given by 1964, the GOP will have been in power twelve years. A one term Nixon is a very real possibility indeed, if he's elected President in 1960.
Maybe; Khrushchev may still shoot himself in the foot with all his
agricultural jiggery-pokery and is vulnerable to the fact that many of the top members of the CPSU - men like Suslov and Brezhnev - are unscrupulous neo-Stalinists.
Not sure why that'd occur at roughly the same time as OTL, given the sheer amount of butterflies unleashed by a '60 Nixon victory...