I doubt that was the reason for the design. The idea that a "warp ring" would actually be a plausible design for an FTL starship wasn't theorized until 2011, after which people retroactively started claiming the Starship/TMP ringship somehow "predicted" it.
Only in the movie. The original novel portrayed a much friendlier relationship between the American and Soviet crew members. Peter Hyams added the Cold War elements to make it more topical for the mid-'80s, but it just made it dated when the USSR fell a few years later.
You are right on 2 good points: Trek did not "create" warp drive, matter/anti-matter drive, not Ion drive, but people keep saying that Trek did. Even the Press gave Trek credit for "creating" the Ion Drive in the Dawn craft. And, I agree that the Soviet military in 2010 was more a creation by Hyams and not what Clarke intending. Hence, the original comment about the military not being an element in 2001, other than the static and unidentified orbiting nukes. Thanks, I appreciate this.
Pew pew" is onomatopoiea for the sound of ray guns firing. So they're talking about sequences of action and combat.