You'll be wanting the 3rd Edition Planar Handbook, whose section on the City of Brass helps flesh out the Plane of Fire as well; the 2nd Edition sourcebook The Inner Planes; and the 2nd Edition Al-Qadim sourcebook Secrets of the Lamp (if you can find it). The 2nd Edition Planescape Monstrous Compendium III offers some notes on Fire wildlife such as the waiveras and scape, but the only new indigenous creature it introduces is the fire bat (other entries such as animentals, fundamentals and ruvoka pertain more broadly to all the Elemental Planes and may or may not be of use). If you want to poke around in it, the 2nd Edition Mystara Monstrous Compendium includes the helion and pyrophor, two more unusual denizens of the plane. Finally, there is a Planescape adventure (The Eternal Boundary) that takes a spin to the plane, but overall it's nothing you wouldn't get coverage of from other sources.
You may want to ask around on Planewalker for more information or check out the work they did on their 3.5 setting conversion; it includes a section detailing the Plane of Fire.
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.
- 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.
The Feywild in 5e has only minor differences from the 4e lore.
The planar cosmology described in D&D 5th edition's Dungeon Master's Guide goes to great lengths to retain some consistency with earlier editions of Dungeons & Dragons. For example, the Feywild (introduced in 4e) still exists, as does 4e's Elemental Chaos, but now the Elemental Chaos is a ring around the separate elemental planes which existed in 3e and earlier.
In other words, D&D 5e intentionally attempts to retain lore compatibility with earlier editions, and this is particularly true for the Feywild, which was an original creation of 4e that was maintained by 5e, and did not need to be changed to accomodate any lore from earlier editions.
The "core" Feywild lore for 5e is described on DMG p.49-50. It is described as a realm of everlasting twilight where the sun never truly sets or rises, inhabited by magical creatures, with civilized lands ruled by the thee seelie fey Queen Titania of the Summer Court and the Queen of Air and Darkness of the unseelie Gloaming Court, existing parallel to the Material Plane, where natural features are exagerrated, which can be entered or left by a fey crossing, and which optionally may cause memory loss or where time may pass at a different rate. This is the limit of 5e's lore on the Feywild.
The 4e books Manual of the Planes and Heroes of the Feywild are largely compatible with this lore, with only minor differences. The Summer Queen is named Tiandra, rather than Titania, and the Gloaming Court is not so explicitly aligned with the unseelie fey. The 4e lore describes a greater variety of terrain, such as lands of permanent winter, and more than two factions among the fey. However, it is otherwise largely the same, and the 4e books greatly expand on the brief description given in the 5e lore.
Aside from this, the multiverse cosmology of D&D 5th edition holds that every DM's game is its own parallel world, and therefore ideas of "canon" only really matter as far as your own campaign world. As per DMG p. 4: