Yeah, the Nazis did find out some valuable stuff about rocketry just because of the amount of money they poured into it but it's greatly exaggerated. Considering that they spent one a half Manhattan Projects on their rocket programs they should have gotten a lot more out of them than they actually did. Goddard taking rocket science as far as he did on a university research budget that was tiny by comparison is a lot more impressive. If we hadn't had any of the Nazis' research I think the USSR and American space programs would have been delayed by maybe two years but not much more.
It's hard to say exactly. There were drivers of the development of the ICBM that were independent of the Nazis--the development of precision gyros for inertial navigation and lightweight fusion bombs for the warhead, for instance. There were designs that were independent of the Nazis--Bossart's Atlas is the preeminent example in the United States. But at the same time the V-2 really was the basis of most post-war work, and most of the people that went on to do most of the work in the later '50s and especially the '60s cut their teeth on looking at V-2s. So it's not clear how much not having the Nazi program would have slowed the development of ICBMs.
That aside...if you want impressive, check out the early Japanese space program, in the 1950s and 1960s. All the way to orbit, as practically a university program. Probably the most economical progression to orbital flight in history!
Von Braun's later role with NASA is also frequently grossly overstated. Initially he wanted to pretty much just toss science aside and blast a rocket at the moon to see what would happen (the "direct ascent" method); lunar orbit rendezvous thankfully won out. He did, to his credit, eventually come around to the LOR way of thinking but still...The primary choice should be the one that needs a bigger rocket than the Saturn? Really?
Saying that he wanted to "toss science aside and just blast a rocket at the moon" is a crude and shamefully inaccurate distortion. At the time, rendezvous and docking
had never been accomplished, and it was far from clear that it would be as easy and routine as we now perceive it to be, after fifty years of work. It was entirely reasonable to conclude that
requiring rendezvous and docking,
especially when failure could mean crew death, was too much risk to take from a technical point of view, especially when avoiding it merely required building a bigger rocket. After all, they were already working on the necessary engines, and everything else is just sheet metal (I'm exaggerating, of course, but that's how it's often perceived by engineers). When it became clear that LOR was a very superior mode, von Braun did quickly switch to supporting it.
That's all aside from the fact that all of the modes were supposed to be basically equivalent in science terms, i.e. pretty minimal (it's not like Apollo 11 did very much at all). The whole goal was to land a man on the Moon, not really to do science. In the slightly longer run, both of them would still have been fairly equivalent for J-class-like missions; direct ascent would have had a larger spacecraft and extra crewman, but would have been hampered by needing to support microgravity and low-gravity life, whereas the LM could be more optimized for just lunar surface operations. Of course, the best mode in the long run--i.e., to create a sustainable program--was EOR/LOR, but that was never considered by anyone, as far as I know.