1. Is there any evidence that Gandalf ever used Narya during the three books? Considering it was rumoured to cause courage in the hearts of men, could one assume that he used it at the great battles like the Pelenor fields to rouse some bravery in the soldiers?

No, I don't think we ever know that he is using it. As DomA suggest, we can surmise that he does.

2. The Ring bearers can see one another's rings. How did Frodo not see Gandalf's, yet he saw Galadriel's? Was he perhaps just not observant?

I think the key is that she let him. He doesn't see Elrond's either. I must say, Sauron must have kind of been a doofus to not be pretty certain that Elrond and Galadriel held 2 of the 3 Elven Rings. I mean, really, who else would have?

3. When Gandalf tells the Balrog that he is the 'servant of the secret fire, keeper of the flame of Anor', is he referring to the Ring?

I always assumed this had something to do with Valinor, not the Ring he bore.

4. Why did Gil-galad give the greatest of the Rings to Elrond? One would think that he'd give it to Galadriel who is far older and of the highest kin amongst the Noldor. Instead she kept the weaker ring while the most potent was given to a half-elf.

I would have to reread to be certain, but a couple reasons off the top of my head: 1) expedience. Did he give it to him right before the battle where he died? Elrond was there, Galadriel presumably was not. 2) Are we sure that Gil-Galad ever had Galadriel's ring? I kind of thought that Celebrimbor gave it to her directly. I could be wrong. 3) Although Galadriel is older than Elrond, I don't agree that she was of a higher kin than Elrond. First, she was not of the High King's house. Finarfin's children were never High Kings, whereas Elrond is a descendant of Turgon and has a claim, should he make it, to be High King of the Noldor. Further, Elrond, as you noted, has human blood, but he is also a descendant of Melian, so his blood is far "nobler" than anyone else, Elf or Human.


5. Why were the Elves not dominated by the Rings like men were? There are several suggestions out there, namely that they took the rings off when they sensed him, or that their will could match his, and finally, that Sauron never tainted the three Elven rings made by Celebrimbor, though he had access to the others.

I think the complete answer to this would probably qualify as a doctoral thesis in Tolkienism, but both things you suggested are part of it, and also incorporate what DomA said.


6. Is there any evidence that the Rings of Power failed when the One Ring was destroyed? As far as I can tell, the Elves were merely paranoid about it and went to the West 'in case' their lands withered.

Yes, I think the Elves power, including the magical powers of the Rings, was destroyed by the destruction of The One Ring. Galadriel was correct about that. The Elves who stayed, like Celeborn, would become less and less magical over time until all their magic completely faded away. Tolkien implied that they would all have to leave Middle Earth eventually, because there was no place for them in the World of Man.