Did do the Heavensward story? if you did, wouldnt need to ask those questions. Well unless you skiped the cutscenes.
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They turn into dragonkin, not actual dragons.
As explained by the shade of Nidhogg before Estinien is possessed, Nidhogg('s hatred) was able to possess him because he was bathed in Nidhogg's power and blood. (Seriously shoulda taken a shower, Estinien! You had plenty of time while we were out in the Hinterlands!) There's also likely some resonance with their personalities in play, but that's unconfirmed.
What about him? His eyes were intact, as was his corporeal body. He used the aether from sealing our Blessing to construct an avatar that followed us around (and snarked at us whenever he felt like it). It's one of those things that got lost with JP!Middy's "chattiness." Regardless, the point is it's the real Midgardsormr that followed us around, or at least used an avatar to act as an observer during 2.5 - 3.0.
Nidhogg, on the other hand, straight-out died. He is dead. We saw his body dissolve into aether. Literally everyone talks about him as if he's now a ghost - Nidhogg has been reduced to a shade sustained by the rancor that lingered in his eyes, possessing the body of a living mortal and somehow regaining his former living shape. Since a dragon's life-force is in their eyes, it's not implausible but without said eyes said dragon will die.
Midgardsormr was just taking nap.
The eye thing applies to every dragon, it's just that most get died before they reach the power of, say, a Celestial Dragon / Great Wyrm.
See above on "dragonkin vs. dragons." True dragons are... well, dragons. Dragonkin, on the other hand, are dragon-like beings such as aevis and diresaurs that are kind of like dragons but not the genuine article. These sorts of things are what Ishgardian heretics transform into upon consumption of dragon blood due to Ratatoskr's (diluted) power in their veins. (Which really doesn't make a whole lot of sense to me, seeing as there were no female knights in either the original Knights Twelve or the Heavens' Ward we fight. But I should pull my mind out of the gutter before it gets into another "How to Au Ra have children?" type thing. Ehh...)
King Thordan and the Knights Twelve ate Ratatoskr's eyes. How? Not sure.
Not sure what happened to Bahamut's eyes either. Presumably, the fact he had none lingering in the physical realm, and the fact Thordan & co. were able to eat Ratatoskr's means they're destructible. Don't need much more explanation than that.
Whose? Bahamut's? It wouldn't make much (if any) difference, since primal-Bahamut was just a pale imitation of the genuine article, not to dissimilar to how Estinien-Nidhogg is just Nidhogg's lingering rancor given physical form.
As for the latter... well, I think it was stated by Hraesvelgr that Ratatoskr's power flows in the blood of the descendants of the Knights Twelve (by now, most every native Ishgardian). Given children only receive half a cell from their father upon conception, it... doesn't make sense for this to be the case? Not going to go further into the somewhat sordid details...
I'm overthinking it, to be honest. Magical space dragons. Or just need to restudy biology. Ehh...
Standard rules of genetics and inheritance don't really apply. By old-fashioned rationale, "blood" is something you get in a 50/50 split from your parents, which would mean dragontaint passes on either way, diluting but never leaving the system. (Seriously though, for all their flying cars and bio-engineering and get-a-load-of-our-singularity space-age bullcrap, even the Allagans operated on antiquated logic where "blood" was concerned.)
And yet, it still makes more sense than anything in Hildibrand's quest line...
3.2 Spoiler, Read only after completing 3.2 or under the condition of giving no f***s about major plot spoilers.
Yea, Eistenian is actually physically fuses to Nidhogg's Eyes. The Dragon can now freely switch between his and Eistenian's form as well. So apparently "Ghost" is a strong word since his Mind was still intact. (His sanity is still missing but it left him a thousand years ago)
3.2 stuff in here
It seems like the obvious solution is to destroy the eyes, which could have interesting ramifications. Lahabrea may or may not be stuck in one of them, which makes Estinidhogg a weird beast for sure.