That was a wyvern egg. Dragons ≠ wyverns.
In context of Fire Emblem, I'm pretty sure you're incorrect. My memory of Sealed Sword and the wiki both have wyverns as being one of many subspecies of dragon, and I definitely remember wyrmslayers working on wyvern riders. Which also means no on the shark thing, at least in context of this manga, since it has wyvern eggs hatching externally.
Which makes one wonder how pregnancy works when switching forms. I don't think the egg would disappear and teleport the fetus back into the womb upon transformation. So it's probably some form of viviparity (aplacental or not).
I mean, they can transform from human to dragon and back within the space of a minute without damaging their clothing or any of the various alchemical salves or enchanted items in their pockets. There is clearly some shenanigans going on here, so as long as the egg fits inside the smaller of the two forms, I'm willing to believe it just works. That said, it's also possible that form switching prior to egg formation induces miscarriage, and human babies are already barely small enough to pass through human hips. There is a nonzero possibility that they have to avoid transforming for the early stages of pregnancy, then transform at the end to actually give birth.
Also birds are dinosaurs.
All birds are dinosaurs, but not all dinosaurs are birds. I don't know how old birds' mode of reproduction is, so while I would imagine all theropods are going to be pretty similar, the other branches might not be. Raptor egg farming yes, brachiosaurus egg farming uncertain.