I have some questions that I can't seem to find the answers for. Any help would be appreciated.

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?
Yes he used it all the time

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?
Galadriel showed Frodo her ring, you could only see it if the ring bearer told you about it.

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?
No, the secret fire is actually the life force that Iluvitar was able to utilize to create life that Morgoth coveted.

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.
Because Elrond was the head of Noldorian houses at the time. Galadriel was actually one of the Sindarin houses after marrying Celeborn.

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.
All those reasons are true. The key being that when they sensed Sauron put the one ring on, they took theirs off. And never put them on again until after Sauron had lost the one ring.

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.
They failed. Tolkein said they lost their power. Tied to the one ring.

Thanks


(I am the scribe, Grimbear gave these answers. Send your complaints to him!)