Compare your own education to what the average citizen of the United States or its territories would have received pre-Civil War; the three Rs (readin', writin', 'rithmetic). Unless you came from money, and were male, that's about all you'd ever get, apart from practical knowledge of a trade, and things every boy should learn like horsemanship and marksmanship. In more civilized regions there was a "secondary school", which for men taught more advanced "ciphering" (math) and what was the current standard in science (remember that in 1850, most of what we now take for granted about light, sound, chemical reactions and the nature of matter hadn't even been discovered yet). For women, that was "finishing school", which was one or two years of etiquette, protocol and social skills (homemaking skills such as cooking, sewing etc were taught at home by their mothers). Only the elite men (both in wealth and intelligence) moved on to college, where they would learn the 50-year-old theory that all matter was made of atoms, and that when chemical reactions occurred the matter was neither created nor destroyed, it just changed form.
I can't speak for your age, but if you've graduated high school, your own education, if typical, would have included algebra, geometry, trigonometry, biology, chemistry, and if you were on more advanced science or math tracks, 2-dimensional calculus, projectile and electromagnetic physics. You take for granted the idea that light is made of photons (first theorized by a young Einstein in the early 1900s), that matter is composed of atoms, which we now know are in turn composed of protons, electrons and neutrons in a centralized structure, and that the movement of electrons causes electricity. The periodic table, the initial forms of which were brand-new in 1850, is now in such a refined form that you don't miss much even if your classroom had a 40-year-old copy of it. None of these things (hopefully) are mysteries to you after receiving a standard government-supplied education.
Now project this another 250 years into the future, to the 2270s (roughly the time of the TOS era). A nuclear war notwithstanding, humans have gained probably another 150 years' worth of development in the sciences, and that might well have represented the same exponential curve we've seen in the last 150 years. Our knowledge also made a quantum leap forward with Zefram Cochrane's theories and subsequent proof of the possibility of FTL travel, which just happened to garner the attention of an alien race passing through the neighborhood. They showed up to welcome us to the space-faring world, and the simple realization that we weren't alone, coupled with knowledge shared by our new extraterrestrial friends, caused a second Renaissance on Earth.
In Kirk's time, things we currently don't even know exist are taught to high-schoolers. Things that are currently the cutting edge of quantum physics being discovered today by post-docs are covered in the equivalent of a freshman science class. Some of what we currently think we know is regarded 150 years from now with the same level of derision as we now view the concept of the four classical elements. Kids win science fairs by building working warp coils (antigravity generators are a relatively poor second place for high school competitions; you learn how to build those by helping your dad tear down his hoverbike).
The pace of education must necessarily match the pace of technology in order for technology to continue to advance. We as humans have to learn what we already know in order to progress toward discovering what we don't. Humans must also adapt, in order to learn more faster. The "Flynn Effect" (the continual, long-standing rise in IQ observed in IQ test scores since the 1930s) has been observed since the formation of the first standardized intelligence tests.
Since we've seen in the Star Trek series a level of technology far surpassing our current knowledge much less our applied capabilities, it stands to reason that the people we see in the series are a combination of all three of your postulates; they're naturally more intelligent, they're better educated, and at least the officers' corps represents the cream of the crop in intellect. Much like in our current educational system, in the future we may identify areas of aptitude and interest in our children and steer them toward more specific areas of study even earlier than we do now (where a "major" of a college Bachelors' degree represents only about the last 2 and a half years of targeted study, out of 15+ years of "general curriculum"). In such cases, those with high aptitude in "STEM" subjects may be groomed for work at Starfleet, which is less like our current military in terms of the average member, and more like a combination of NASA, Microsoft and Lockheed; in short, you want to work there, but you have to be damn good to get the chance.
And just like in our world, the guys who goofed off in Physics will manage the physicists and get all the glory (you think I'm joking, but if I'm wrong, why isn't Sulu captain? He's just as good as Kirk in just about everything we've seen them both do, and without a blemish on his record, but he's second officer at best).