People blindly asserted that Twilight hatching the egg was some rare and unusual event. There is no indication of that. There is every indication that using dragon eggs to test students applying at Celestia's school is a standard yawn-worthy affair of no special significance.
Where? All I can recall was my nonserious remark that it was some sort of false-front exam. The consensus seems to be that they were shocked because she grew Spike into a mature dragon (or a parody of one), not that she hatched him. I don't know where you are pulling this from.
the master is still taking care of the familiar. It's not slavery.
So if I kidnap your baby and task him with menial labor...but take good care of him, that's not slavery?
You assume that the impression is against the will of the parent dragons, yet there is no evidence for this. Given the nature of Equestrian society and that Celestia doesn't have the time or energy to constantly fight off suicidally angry dragons, it is much more likely that they are involved in some sort of mutualistic relationship where eggs are sent to the academy to be raised, strengthening the bonds between dragonkind and ponykind, eventually outliving their partners and joining the dragon population as adults. You could even take it as being an education for them, not only because of what they will learn as the companion of the equivalent of a grad student, but the lessons on humility, acceptance, friendship, etc. Again, see Pern for a perfect example of a mutualistic relationship between two sentient species, one of which happens to be dragons.
They've been in Canterlot for the Grand Galloping Gala, and one would have expected to see at least a few graduates from such a prestigious school there
Spike didn't go. Why assume that other students would bring their familiar? And Spike also didn't go when they went to Canterlot in The Return of Harmony.
You don't take your slaves with you to the palace.
In reality, slaves are actually one of the classes with the greatest access to the wealthy and powerful, as they are used in a multitude of tasks. If Spike were a slave, it would have been much more fitting for his "master" to bring him along to take care of problems beneath her notice.
The evidence, such that it is, supports the idea of a mutualistic relationship where dragons send their young to be educated in areas applicable to them (mainly to reduce friction in intracommunity interaction between dragons, as well as how they interact with other sentient species), essentially for free. Keep in mind that the dragons will outlive their unicorn partner by a great span of time; the years that Spike lives with Twilight will basically be the equivalent of birth to puberty, assuming ponies have a lifespan of 60-100 years. By the time the unicorn has died, the dragon will be old enough to strike out alone and begin accumulating a hoard. If, as I believe, this system has been in place for centuries or millenia, it has been very successful at fostering understanding and harmony between ponies and dragons, to the point that dragons both apparently avoid eating ponies, and are actually willing to accomodate their needs, as in Dragonshy, once they understand that they are causing trouble.
This in turn benefits the dragons by removing the responsibility from mothers to split attention between young and (to dragons) more serious concerns, such as protecting their treasure, as well as heading off conflict between the two species at large because some foolish dragon child pissed off Celestia by eating a pony, burning a town, or some such thing. In short, it would promote the exact same values as the show itself: friendship, harmony, and acceptance.
Q: "Well, why are there no other known dragon babies being raised by unicorns?"
A: Dragons are sort of hybrid breeders, possessing the K-strategy trait of producing few viable offspring, yet also the R-strategy trait of not particularly caring for their well-being (according to most canon interpretations of dragons). Likely there are dozens of new students each year, but there are not enough eggs. So you test each applicant with the egg, and if they don't hatch it, they proceed normally, but if they do, they become the caretaker for the baby dragon. Thus, you have the hatching/partnership as a viable system without having baby dragons all over the place.
Q: "If Spike isn't a slave, why does he do chore and why does Twilight not take him absolutely everywhere with her?"
A: Because he is a child, and Twilight is a surrogate mother. Parents give children chores to teach responsibility and a work ethic. Twilight gives him what is occasionally a bit much to do because she is an academic and as such is a bit detached from the physical world when she is pursuing an idea. Not the best, but certainly not "slavery". Misinterpreting their relationship is rather difficult to do unless you are deliberately doing so/have never had parents, siblings, or children. Or caretakers at all. As for why she doesn't take him everywhere... Really? He is a child, and she is essentially a grad student conducting field research. Not only would much of it bore him to tears, but she often gets into dangerous situations, and does her best to keep him out of danger, as part of her role in raising him. Obviously she fails, because being inattentive isn't the best trait around a curious child. Herpaderp.
And again, I need to emphasize this:
IF CELESTIA WAS KIDNAPPING DRAGON EGGS, SHE WOULD HAVE ANGRY DRAGONS, BURNING TOWNS, AND MANGLED PONIES OUT THE WAZOO. SHE DOES NOT, SO CLEARLY SHE ISN'T KIDNAPPING THEM. Dragons may not care much for their young, but eggs are another kind of special, shiny treasure to them. They would retaliate in a rather nasty fashion if they were stolen.
Regarding the images: Animators putting in scenes that are mildly sexual =/= canon child slavery.
Rarity and the Diamond Dogs: Is Rarity a child now? Or a dragon, perhaps? Are Diamond Dogs ponies? No on all counts. The fact that a trio of individuals from a species that has been portrayed as much less objectively "good" than ponies abducted an adult with a special talent does not imply that Equestria is home to the systematic kidnapping and enslavement of the unborn children of giant firebreathing lizards. Sort of like how your first image of some ponies in compromising positions while fighting under the influence of a love spell doesn't imply that ponies across Equestria have wild sex orgies every Thursday night.
Sorry for the great wall of child slavery.