Nixon, even lacking his trademark paranoia, wouldn't have been ready to be President at this point. From what I understand, Eisenhower didn't think very much of him during their time together as President & Vice President, & I think that if Ike had succumbed to such a freak accident, then the electorate would agree. I think many would also be quite upset that the man they idolized for his service in WW2 was replaced by such a man as "Tricky Dick."
There's also the fact that Nixon was rather close to McCarthy, & while he was no idiot, I think he might indulge the man more than Eisenhower did, which could be very ugly for him.
As far as foreign policy goes, I don't have much to base this on, but I do get the feeling that Nixon would be far more aggressive (though not as competent) in responding to the Hungarian Revolution & Suez Crisis, which would be a disaster, in all likelihood.
I'm not sure what would happen in 1956, to be honest, but I do get the feeling that the Democrat (be it Adlai or somebody else entirely) would win.
I agree. I think Nixon would have have been a much better president if he was elected in 1960 instead of 1968. Before the 1960 loss, he was far less paranoid, but still had his great political instincts. It's hard to deny Nixon was a political mastermind and knew how government and international affairs worked. Furthermore, he was more moderate in these days considering the Southern Strategy wasn't a thing back then.