Short answers for why the eagles didn't fly the ring to Mount Doom.
The eagles are not mere beasts of burden; they are the divinity, or close to it. They are the personal servants of Manwe Sulimo, who is the chief of the Valar (archangels, basically), and are therefore basically equivalent to Gandalf - himself a divine being - in terms of authority. They would probably have viewed being used as a personal Hobbit taxi service to be an insult.
Remember how Gandalf didn't take the ring because he thought its power would corrupt him? The eagles are no more immune to that corruption and they themselves almost assuredly know it. They would probably refuse to bear the ring for the same reasons Gandalf did. One of the reasons why Hobbits were used to carry the ring is that they are remarkably resistant to its corrupting influence (the ring plays off the desires and insecurities of anyone nearby, basically promising them whatever they want in return for giving in to its power; since all most Hobbits want is a strong pint of ale, some weed to smoke, some earth to till, and a peaceful, quiet life, there's nothing the ring can offer them that they don't already have).
The whole point of sending a small group of heroes to convey the ring as opposed to, y'know, a massive army was that this was a stealth mission. Sauron was scouring the land for any sign of the ring, and given that Saruman was sending birds on patrol to watch key passages the skies were certainly not safe. If the Council of Elrond had sent the ringbearer to Mordor on the back of an eagle, there would be a bunch of ringwraiths, ballistae, and guards with bows and arrows waiting for them when they got there.
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Kill 1 man: You are a murderer. Kill 10 men: You are a monster. Kill 100 men: You are a hero. Kill 10,000 men, you are a conqueror!