Some comments from Aaron led me to do some more thinking about Dragons and to articulate my views.
In a nutshell, I think of Dragons as huge, long-lived, essentially solitary predators who happen to be very smart. And I wondered what sort of civilization such creatures would develop, because their interactions with each other would be limited and they would so often be in competition with each other. Plus you could only have so many of them, particularly Elder Dragons.
I saw Draconic civilization as progressing very slowly because individual insights were so rarely shared and they would create very little infrastructure. One big Elder Dragon may be very smart and very powerful, but how much can it really achieve on its own in terms of creating infrastructure? Especially if it has to guard its territory against incursions by other big, powerful predators?
To me this screams out for servants. No Dragon could successfully raise herds of food animals, for example, because they couldn't control a bunch of beasts terrified by having a huge freakin' Dragon around.
But human servants could. Human servants can build great works over long periods of time and produce large amounts of mundane goods that would be wasteful and boring for a powerful Dragon to spend the time making. Human servants acting as proxies could engage in tentative cultural and economic exchanges with other Dragons who by their nature would be uncomfortable and possily violent if gathered together. Human servants can even explore the natural world more efficiently than a single powerful Dragon, traveling around in groups to collect specimens, conduct research, and compile knowledge. Human servants are also the best tools for hunting and controlling groups of "wild" humans that might encroach on a Dragon's territory. Humans can even help defend a Dragon against its rivals, acting as an early warning system.
The Dragons that "domesticated" humans thus gained advantages similar to those acquired by humans who domesticated animals like horses, cattle, dogs, and sheep: extra labor, expanded and reliable food supplies, protection, and greater access to goods. In essence, Dragons paired with humans led to accelerated civilization for Dragons.
The Dragons that followed this path out-competed their rivals. Then they set up a social system that controls Dragon populations in settled areas (by limiting reproductive rights for Dragons, promoting wars of conquest, and limiting the number of Elder Dragons active in any area), selects for the strongest (by sterilizing those with the least ability), and provides outlets for the instinctive predatory desire to acquire territory (by tying active Dragons to large feudal fiefdoms and encouraging constant exploration and expansion).
At the same time, the Dragons that bond with the human nobles of the Great Houses often forge strong emotional ties with certain lineages. Think of it as a curmudgeonly farmer who has few social contacts but who breeds and trains sheepdogs. That farmer grows to have real pride and even love for his animals, which are more than pets--they are work and life companions.
Now, all of this is colored by the fact that humans are more intelligent relative to Dragons than cows, horses, and even dogs are relative to people (though you might not agree!), with the most obvious example being that Dragons and people can communicate with each other using sophisticated language. But the underlying dynamic is still similar to that of domestication and pets/working animals. Some Dragons will treat humans as disposable, while others will love their closest human companions more than they are capable of loving any of their Dragon peers. But they'll all see humans (and the other intelligent hominids) as inherently useful.
As to Dragon frequency, Elder Dragons are rare and mysterious in my setting, their motives uncertain and often frightening simply because they take a very long view. Adult Dragons are rather more common, but not in groups--there tends to be one in an entire principality, for example. The exceptions are the sterilized Dragons, who are much smaller but still long-lived and secretive, tending to work behind the scenes to shape human civilization.
Juvenile Dragons are still rare enough that most people would only see them from a distance, but they are often assembled into Flights that serve the Empire as a kind of aerial shock troops. Those that prove themselves and survive for 20 years or so of service can gain reproductive rights. This is the most socialization that a typical Dragon will ever have.
Ironically, when other cultures write about someone slaying a Dragon in heroic battle, it is almost always a juvenile Dragon that they are talking about.