Interesting discussion... several thoughts. To be frank... I think "then why did the ring pass right under Sauron's eye in Dol Guldor, in Mirkwood, with nary a blink?" was because Tolkien had not fleshed out LOTRs yet and so did not consider that in the writing of Bilbo's adventure.
On Amen Hen... Frodo was more visible to the Eye for two reasons... one, as stated, he was on the Seat of Seeing, but the other was because Frodo had suffered the Morgol Knife and so was forever partly in the wraith world, or at least more visible to it. Also, Frodo had grown larger in a way... more than Sam.
I agree that those with power and those that desired it were more easily corruptible by the Ring, but they would also have had more power to hide themselves from the Eye. (Gandalf or Galadriel)
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