Your emphasis is on fey, but it’s actually easier to start with fiends and celestials.

Also, this is going to necessarily be based on the history of D&D, because 5e hasn’t really gone into a lot of details about this sort of thing. That said, everything I claim here is consistent with 5e, including the things that they have changed.

OK, so then, fiends and celestials are what previous editions of D&D called “Outsiders,” non-mortal creatures from other planes of existence. They are related to “Elementals,” and share many similarities with them—one could think of Outsiders as being the “Elementals” of non-elemental things, most notably belief. That is, where a fire elemental is a being made of fire, “fire incarnate,” a fiend is a being made of evil, “evil incarnate.” Celestials likewise but good. In the wider multiverse of D&D, most thoroughly described in the Planescape setting of 2e and 3e, belief in alignments is potent stuff, giving rise to entire planes of existence (the heavens and the hells and so on), which are made out of solid belief in that alignment. Fiends and celestials are made out of that same solid belief as the plane they originate upon—and since those planes of belief are known as the “Outer Planes,” they are known as Outsiders (for the record, the “Inner Planes” would be the elemental ones, but no one calls elementals “Insiders”).

One of the key things about Outsiders and Elementals both is that they do not exhibit “dualism,” the concept of a soul and a body as separate entities. For an Outsider or Elemental, their soul is their body and vice-versa. This allows their body to radically change in tune with changes to their soul—since those are the same. For instance, a marilith, a six-armed demon with the lower body of a snake and the upper body of a woman, could become a balor, a hulking, furry brute with horns, cloven hooves, and enormous wings.

^{A balor and a marilith, as depicted in the 3.5e Monster Manual.}

Likewise, they could be bound to other forms, say objects. They could inhabit other creatures, possessing them. And so on. This is how fiendish or celestial “spirits” can be used as a familiar or steed.

How do Fey work into this? In previous editions, they didn’t; this use of “fey spirits” is new to 5e. However, some Fey creatures were incorporeal spirits—no body to speak of. Unlike, say, a satyr or dryad, some fey were only spirits—making them more like Outsiders or Elementals. And the Fey were largely associated not with the elemental planes, or the planes of belief, but with the material plane¹—the plane of mortals. They are, in some ways, analogous to Outsiders and Elementals for the material plane. And this is exactly how 5e uses them: where in previous editions, a hag was a “native outsider,” that is an Outsider of the material plane, in 5e hags are classified as Fey creatures instead.

So 5e changed Fey to be more like Outsiders, and so it would seem this included the greater malleability of Outsiders, to allow them to be bound into conventional animal forms.

1. _{Or the Feywild, which was a new plane introduced in 4e that 5e has retained. But since the Feywild is “an echo of the material plane,” it still associates Fey far more strongly with the material than either the Elementals or Outsiders would be.}

Yes; a shape-changed dragon can be cloned, and retain the ability to change back into a dragon.

When a dragon changes shape, its human form retains the ability to change shape (MONSTER MANUAL, p114). Therefore it would follow that a clone of the human form also has that ability. The Clone spell description states that the clone has the same abilities as the original (PLAYER'S HANDBOOK, p222).

An option for a possible adventure hook, if the above doesn't suit your game, would be the human clone of the dragon recruiting a group of adventurers to recover its remains. This would allow the clone to be slain, freeing its soul, and the dragon's original form to be raised from the dead or resurrected.