The Eagles are not normal creatures. They are scions of the Second World, that of Valinor. Their ancestor was an angelic spirit, a Maia, that took the form of a giant eagle in the Days before Days, in much the same way that the spiders of Mirkwood and Shelob the Great were children of Ungoliant, a massive spider-Maia that dwelt in Valinor before the Days of the Sun and Moon and that helped Morgoth poison the Two Trees and steal the Silmarils.
Maia that dwell in Middle-Earth at the time of the War of the Ring are Saruman, Gandalf, Radagast, Tom Bombadil, the Balrog, Sauron, and possibly the Watcher in the Water, although I believe that it was another descendant of a Maia, one of Ulmo's spirits that took the form of a water-creature like unto a great octopoid.
The Eagles can talk and are quite intelligent. They are the eyes of Manwe, the leader of the Valar, who are archangelic spirits that dwell in Valinor and watch over Middle-Earth from afar. It was Manwe who sent the Istari, or the Wizards, to Middle-Earth to inspire the Free Peoples to resist Sauron. Like the Istari, the Eagles are prohibited from interfering directly, and only do so in order to rescue or to help in the direst times - and only then when the Free Peoples have given it their all.
They rescued the dwarves, Bilbo, and Gandalf from the goblins and wargs, but did so only because of their allegiance to Gandalf, who healed their chief from an arrow-wound years earlier, and because they knew he was a Maia.
They participated in the Battle of Five Armies only after the Free Peoples had given it their all and were still losing, and again because Gandalf was present.
Gwaihihr rescued Gandalf from the Tower of Orthanc again for the same reasons he rescued him from the goblins and wargs.
The Eagles participated in the battle in front of the Gates of Mordor for the same reasons they fought in the Battle of Five Armies, and again because Gandalf was present.
So, why didn't Gandalf try to talk the Eagles into flying the Ring to Mount Doom? Because he knew the Eagles would refuse - the Ring was not their burden to bear - it was of Middle-Earth and must be destroyed by those of Middle-Earth. Their mandate from Manwe is one of non-interference.
Edit: the "sulphur" clouds of Mount Doom weren't the issue. It was the fact that the Ringbearer would have had to have landed and then made his way into the Mountain. According to book canon, it took days for Frodo to find the Crack of Doom inside - it wasn't just a "run in the door and toss the ring" kinda trick. By the time he found the correct place the Nazgul would have been on him, as they know the Mountain much better than Frodo could.
Sauron was never described in the book as being an Eye perched visibly on the top of the Barad-Dur (as he was in the movie), he was still yet spiritual in nature, a maleficent Presence that filled the Barad-Dur, unsleeping and always watching. Sauron could see into the shadow world of the spirit, which is why he almost spotted Frodo on the Seat of Seeing on Amon Hen at the Falls of Rauros - Frodo was wearing the Ring at the time, and why he was instantly aware of Frodo when Frodo claimed the Ring inside Mount Doom.
The Nazgul's Fell Beasts were no match for the Eagles in the air, nor were there any "evil things" left in the world that could have challenged them - Smaug the Golden was the last of the great winged dragons to come out of the Withered Heath, far to the North, and Sauron didn't have the time to try to seek out any possible others that might or might not be alive. During the First Age the Eagles joined the Host of the Valar against Morgoth, and they helped Earendil slay Ancalagon the Black, a dragon so mighty that his fall broke a mountain chain.
Edit 2: The Ring's very existence kept Sauron in Middle-Earth. Even without it his armies would have eventually swarmed over Middle-Earth. Had it been buried in the Shire it would have been found when the Shire was eventually overrun and Sauron came there himself. Had it been left in Bombadil's care then as it said in the Council of Elrond, "Bombadil would fall, Last as he was First, and then Night would come." Had it been cast in the Sea it would have came to Sauron's attention eventually. There are things in the depths that would have found the Ring and they would have become slaves to the Ring, bringing it to Sauron or as close as they could.
The point was that Sauron didn't need the Ring to conquer Middle-Earth - his armies were already too large for Gondor to fight off - in fact, had the Ring not been destroyed when it was Aragorn's army would have been wiped out.