Bartman is spot on here. The Eagles may actually constitute a menace so far as the Ring is concerned. They may or may not be Maiar; Tolkien never says. They were created by Manwe to be his servants, since Manwe controls the airs and winds of Arda, and he needs the Eagles to be his eyes in Middle-Earth and report back to him on various doings. Their allegiance is to him, nobody else.
About Yavanna- she has nothing to do with the Eagles. She may be responsible for the creation of Ents, the Shepherds of the Trees, since she’s concerned that the Children (elves and men) will just wantonly slaughter all her creatures. Tolkien never comes right out and says this, but it’s certainly implied. Yavanna gets the idea to make (or ask to be allowed to make) Ents after she hears how forgiving Eru was for Aule’s jumping the gun and creating the dwarves. At one point in the book, Yavanna mentions that her trees will be tall and strong enough for the Eagles of Manwe to nest in them, to which Manwe replies that Eagles will only nest in mountains, not in trees. Is this the source of confusion, maybe? But Yavanna has nothing to do with the Eagles- they only answer to Manwe.
Now, the Eagles basically couldn’t care less what happens to elves, men, dwarves, etc. The only reason they help anyone at all is because of Gandalf- maybe because he’s their friend, maybe because they know he’s a Maiar and thus they’re working for the same guy. Who knows? Otherwise they just mind their own business. If they got hold of the Ring, things could quickly go downhill. Even though they’re good, we all know what the Ring can do to people who wield it. An Eagle wielding the One Ring would most likely destroy the Children in the name of preserving nature. Again, who knows?
No, Eagles could not just fly to the Ring to Mt. Doom. They could not just fly somebody carrying the Ring to Mt. Doom. The quest succeeded by stealth alone, as has been pointed out a near infinite number of times now. A big freakin’ Eagle with a 30-ft. wing span is gonna look a little suspicious crossing Sauron’s airspace, dontcha think?
Anyway, let’s play along with this hypothesis for a second. Let’s say the Eagles were all good, and incorruptible, and they agreed to carry the Ring to the fire. This will take, what a few days? Great! The Ring will be destroyed in one quick, tidy step. Now everyone’s problems are over. Perfect. But it would make for some pretty fucking dismal entertainment reading, no?
Elrond: So, Gwaihir, back from Orodruin so soon?
Gwaihir: Yep.
Elrond: Destroyed the Ring then?
Gwaihir: Yep.
Elrond: Slammin’. Let’s go get a taco.
The whole point of the books is the quest- the various trials and tribulations the characters go through to complete their mission, and the invaluable life lessons they learn and changes they bring about. I mean, really- it’s just how the book is written. Why not ask why Gandalf didn’t just call down an airstrike at Helm’s Deep? Why not ask why Frodo didn’t just say “Fuck this!” and stay home? Tolkien’s whole point was that sometimes you just have to do things the hard way.