The Soviet equivalent of Operation Paperclip was Operation Osoaviakhim, where the Soviet Union rounded up 2000 German scientists, engineers, and technicians in the Eastern Zone and “volunteered” them to work in the Soviet Union. and at the same time grabbed all the precision machinery and technical data they could, including stripping the base at Nordhausen.
Most of the high-level V2 project scientists left with von Braun to surrender to the US/UK (I think Gröttrup was the only exception). And they also carried off much of the V2 documentation and plans. So while the Soviet’s also carried off V2 personnel, it was largely lower level engineers whose knowledge of the project as a whole was pretty limited. They never played anywhere near as important a role in Soviet missile development as von Braun and co. would play in the US.
The Soviets brought back from Germany some V-2 rockets and parts and a lot of second and third tier engineers. The engineers were there to disassemble, measure, and make quality drawings of the German V-2s and their components. Reverse engineering was the Soviet’s métier at that time.
To return to the original question for a moment, спутник - according to the Russian edition of Wiktionary - means “traveling companion” and “satellite”. Etymologically, it’s derived from путь, the Russian word for “road” or “path”, and the preposition с, meaning (in this case) ‘along with’ - so, literally, someone who’s on the road along with you.
Unfortunately, the online etymological dictionaries I’ve found don’t provide OED-style dated references for uses of the word, but спутник is used to mean “satellite” in the classical astronomical sense - as in the Moon is a satellite of Earth, Charon is a satellite of Pluto, and so on. I would think it likely, then, that it was used in that sense long before 1957 - so seeing as how the Russians actually built a satellite, then by Marx that’s what they were gonna call it.
They had some skilled people. In addition to Gröttrup, there was Kurt Magnus, who was pretty famous for his work with gyroscopes, Joachim Umpfenbach, Johannes Hoch, Werner Albring, and others. That they didn’t play as important a role in Soviet missile development as von Braun did in the US was due more to politics than ability.
Sputnik was also the name of the state tourism board in Yugoslavia until its breakup.
There’s a fairly famous picture of Von Braun, with an arm in a cast, saluting the American forces.
I have heard a theory (almost certainly apocryphal) that he actually wanted to go the Russians, and his arm was broken in a tug of war between the Russian and American forces. The official story was that it was a car accident just before the surrender.
That’s so unlikely I’m sure you must have heard it wrong.
IOW, it’s apocryphal awry.
Remember, also, that when Sputnik went up, many many people were scared to death. This was proof that the Ruskies were up to no good, that we were really threatened by this remarkable event, that the cold war was about to really heat up, etc. It exacerbated a lot of fear that the public felt which certainly helped to energize the rush to scientific activity. (I was just an adolescent at the time, and a science fiction geek, so I was absolutely thrilled when I heard the news. It was so exciting that humans had actually succeeded in putting something in orbit in space. Imagine my surprise when I discovered that just about everybody else was freaking out.)
Roger Kaputnik?
You’re not seriously imagining Russian soldiers literally yanking on one arm and Americans on the other, are you? 
Anyway, the story isn’t true. Von Braun broke his arm in an auto accident.
I think it was an issue of capabilities not intentions. Anyone who didn’t already suspect the Soviets were up to no good by 1957 had to be pretty much committed to a belief in Soviet good will.
But the capability of the Soviet Union to act was still a question. Americans got the first shock in 1949 when the Soviets exploded their own atomic weapon and demonstrated they could catch up with the United States in an important new military technology. It happened again in 1953 when the Soviets exploded a fusion bomb - this time only nine months behind the United States. But at least in both of these cases, the United States could reassure itself that it had gotten there first and the Soviets were only catching up.
Then came Sputnik in 1957 and this time the Soviets were in the lead. They had launched a satellite into space before the United States. The Sputnik launch scared Americans because it destroyed our belief that we were setting the pace on the development of new technology. If we developed a new weapon and the Soviets then caught up with us, equilibrium was maintained. If the Soviets developed a new weapon that we didn’t have, then we might be vulnerable to a Soviet attack we couldn’t respond to.