A wild, faerie plane?
The descendents of dragons?
The plane where dead souls go?
The fey inhabitants of the Feywild?
1. Feywild isn't too bad. But something like The Primal, or Sylvania, or Arborea, or the Spirit World would have more resonance.
2. What do they call themselves (e.g.: what's draconic for them?)? Perhaps call them the Kujuta (what Bahamut "supports" in myth), or Bameh (a corruption of Bahamut), or the Athine (from Leviathan, the monster who represents the "Tiamat" role in D&D in Jewish myth). Or even combine the words, perhaps Batia, or Tiathine. Bump a few letters around to make it flow in a pronouncable way, you've got something evocative and useful.
3. The Underworld. Hades. The Dark. The Afterlife. Sheol. The Pit. Much more resonance. Shadowfell is middling.
4. Eladrin isn't too bad. Fey? Fairies? The High Folk? The People of Light? Sidhe?
Why are you so picky about names?
Whenever I see the word "dragonborn" it makes me think of a dragon giving live birth to a person.
That's not a mental image I want.
"Feywild" works okay, mental-image-speaking, because it makes me think of thick vegetation and a pixie who is very angry. Shadowfell works okay, because it makes me think of a dark pit. But in both cases, we have such a richness of real-world myth to draw from, why would be bother re-naming it something else? Or if we're going to name it something else, why not transcribe it into Elvish or Draconic or Gnomish or Dwarvish or something.
What's the Eladrin's name for their home? What do the dragonborn call themselves? What do the ghosts speak of the land from which they come?
"Feywild" doesn't sound like a high-elvish term. "Dragonborn" doesn't sound draconic at all. "Shadowfell" doesn't sound like the whispers of a half-living spirit.