Taking these one at a time:
What would the soveriegn(sic) dragons think of the tamed dragons
Some of this would certainly come down to how dragons were tamed in the first place. Was it through force or subterfuge--that is, were dragons physically beaten into submission or stolen from their families? (This raises an additional question: Do dragons form families so that theft of eggs or hatchlings to tame would be a disruption of that family structure?) Was it through some kind of mutual protection pact between a small subset of dragons and the human kingdom? Was it by adoption of orphaned hatchlings?
Perhaps think of it this way: Would sovereign dragons see tamed dragons as slaves (taken involuntarily and forced to serve), servants (voluntarily invited into human society but relegated to a lower position in the social hierarchy), or would-be humans (full members of society)? Then shade that with how sovereign dragons think of themselves--their power and dominion over the world--relative to humans. If humans are terrifying rivals who have demonstrated the power to kill dragons in large numbers, and some dragons go to serve them willingly, then the sovereign dragons might regard those tame dragons as quislings and traitors while still emotionally understanding the decision because humans are much more powerful. Conversely, if humans are considered only slightly better than other sorts of animals and aren't much threat to adult dragons, but they've nevertheless managed to make dragon slaves, then tame dragons might be objects of pity or derision for being so weak as to be enslaved. (It seems from the situation you've described that the second view of humans is more likely--but there is always the option that in-universe, the dragons don't know that humans can't actually bring all their military might to bear on them. They may not have a good understanding of human politics, and assume the human empire is a single mighty force still capable of wiping them right off the map.)
How the two populations relate to each other has a lot to do with how tame dragons got where they are, and how humans and sovereign dragons relate to each other. It also has to do with the psychology of the two groups, and how they see their place in the world.
What would the soveriegn(sic) dragons think of the dragon knights
This would also flow out of the particular history of how dragons came to be tamed. If being a tame dragon is just a viable alternative life choice for a dragon, then how sovereign dragons (as a population) feel about dragon knights might be neutral ("oh it's that's Ancaladon, he's got a weird human friend, there she is now") to positive ("Smaug is so lucky, he's convinced a human to bring him food and polish his scales"). If tame dragons are the nth-removed, "enslaved" offspring of kidnapped dragons, dragon knights would be viewed with extreme negativity, as the modern heirs to an age-old offense against dragonkind.
And so on, and so forth. Again, how the populations relate depends a great deal on their psychology and their history with each other.
What would the other kingdoms think of the kingdom with the dragon knights
That depends. As a military force, how practical are the dragon knights to field? How much of a force multiplier are they? That's to say, is a dragon knight capable of taking out an entire opposing troop of foot soldiers--or an entire opposing company? Or are they maybe just slightly better than standard ground cavalry?
If they're a large, scary military force that's only been kept from pushing over all the surrounding kingdoms by not having quite enough dragons to fight everyone at once, it would be reasonable to say that even their allied neighbors are wary of them. If they're just a little better than regular knights mounted on horses and they require a lot of extra care and feeding that includes their dragons sometimes terrorizing the countryside, then their kingdom might be held in contempt for having such a liability as dragons around.
What would a serf think of the dragon knights
What would a serf know of the dragon knights? Are they praised in story and song throughout the kingdom for their daring-do and noble acts? Or are they mostly known in rumor for bullying the common folk and taking whatever they want? I think you'd find a serf's natural reaction flowing from what they know about the dragon knights' behavior, whether it be respect or fear or contempt.
Okay, let's add some new stuff now that my initial question-bolus has been answered!
Sovereign dragons on tame dragons
Whatever else the sovereign dragons think about tame dragons, they're going to think on some level that tame dragons are weird for wanting to live in really close contact with another species and subjecting themselves to laws never meant for dragons in the first place. It doesn't matter that the tame dragons might be getting a really good deal--ready access to food without the danger of the hunt, no threat of predators stealing their eggs and hatchlings, access to all kinds of wonderful human goodies like fine metalwork and gems--they're living in a kingdom that isn't ruled by a dragon. And, unless the laws of heredity are weird enough in the dragon-human kingdom that it's possible for a dragon to eventually ascend to the throne (maybe by adoption into the royal family), they're always going to be ruled by humans. Tame dragons have effectively tossed out some of their self-determination, though a tame dragon might fire back on that one by pointing out that the average low-rank dragon in a dragon-ruled kingdom isn't doing much self-determining, either!
Add to that that humans and dragons are very, very different--physiologically, and undoubtedly psychologically as well. Even without any experience of doing so, sovereign dragons can probably imagine just how strange and uncomfortable it would be to cohabitate with humans: Their houses are tiny, their sense of smell is so weak that they can put up with intolerable levels of stench, and they build with the expectation that anyone using their stuff can balance easily on two feet and has opposable thumbs. And, they don't use any of the same body language dragons do (except maybe the broadest, most understandable gestures, like cowering down to show fear or submission), their voices are all in the wrong pitch, they don't understand when they've said something insulting and they take offense at the weirdest stuff... Think of it sort of liking moving into a neighborhood where everyone's from a totally different culture and even though they speak your language, everything else about them is so alien that you have a hard time fitting in.
Another thing to consider that hasn't been touched on is that the tame dragons will undoubtedly be able to tell their human neighbors everything they need to know about dragon psychology, dragon social hierarchy, the general disposition of the dragon kingdoms (although this news might be several generations old, depending on how much flow there is between the tame dragon and sovereign dragon populations), dragon physiology... Basically if there's a potential weakness in the dragon species as a whole, or the sovereign dragon kingdoms themselves, the tame dragons will know enough to pass that on. And unless they've all sworn a vow of total secrecy on ever talking about dragon inside baseball state secrets (potentially possible) or dragon physiology (not really possible, since dragon cadets will need proper medical help from time to time), the tame dragons will talk about these things. Moreover, depending on how some of them feel about sovereign dragons, they might be all too happy to talk about these things, and try to encourage their human neighbors to maybe do a little counter-raiding of the dragon kingdoms to get some of that sweet loot piled up in those mountains...
The sovereign dragons will undoubtedly realize this. They may even have the offspring of tame dragons who just couldn't hack it with humans showing up in their kingdoms periodically, which means they'd be getting infrequent reports on just how much the human-dragon kingdom knows about them. They'd have reasons to be very worried about that, because it means if the human-dragon kingdom manages to take over the entire human empire, they're in prime position to take over the dragon kingdoms next.
Also, to build off something I mentioned in one of your other questions on this topic (how other humans view the human-dragon kingdom), the sovereign dragons might know very well that the tame dragons aren't the puppetmasters of the human-dragon kingdom: Dragons would never organize a kingdom that way. (For whatever value of "that way" you want to put in there--not enough due deference to dragons, no horrific blood sports for positions of leadership, etc.)
Sovereign dragons on dragon knights
While dragons associating voluntarily with humans is questionable behavior at best, humans who voluntarily associate with dragons are just showing good sense! Dragons are, after all, superior at war and hunting both, and the dragon of the dragon knight partnership makes up for the fact the knight is small, squashy, and not very frightening. At least, that's one way they could think about it, through the lens of draconic arrogance.
But there's other ways. Dragon knights ride dragons into battle, and even if they're not actually commanding the dragon where to go, there's an element of servility in letting people ride around on you that the sovereign dragons wouldn't fail to miss. That bespeaks a certain arrogance on the part of the knights--they think they're good enough to make a dragon into a beast of burden--that would be intolerable to some sovereign dragons. Even ones who have nothing against dragon-human cohabitation per se might have a lot against humans riding on dragons, because it strikes them as demeaning to the entire idea of being a dragon.
Dragon knights are also in a position to know the most about what makes dragons tick. And that feeds into one of my last points in the previous section: Sovereign dragons would know about this, and know that it makes the dragon knights a much bigger danger to their sovereignty than humans who've had only infrequent contact with dragons. Even dedicated dragonslayers wouldn't know as much as a dragon knight does about dragons, and that's dangerous.
