re: Laundry list of LOTR questions

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Nazgul and the Eye of Sauron would strike them down.

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This. And it was established the Hobbit that they aren't just random beasts of burden to be ridden whenever like a steed. They were immensely proud and powerful creatures that had to be shown a lot of deference by someone the stature of Gandalf.

Also in the Hobbit the Eagles made it known that they didn't even like being threatened by the long bows of men protecting their sheep. They aren't all invincible beings that are immune to sharp objects.