Unfinished Tales has a quote, something like "it added to the warmth of the already great fire in his soul", or something to that effect.
Gandalf was a spirit of fire who stayed loyal to Iluvatar and did not defect to Morgoth. So his abilities with fire that he showed could be from just that. But I do remember reading somewhere (maybe History of Middle Earth, though it has been a long time) that his abilities with fireworks, especially, was increased by Narya.
But there are one or two specific examples of its use. Gandalf says that when Frodo nearly used the Ring in the Barrow Downs, he knew. I'm fairly certain this is tied to Narya. Again, when he makes Frodo take the Ring off at the Aragonath at the end of FotR, he could conceivably have been using the power of Narya.

Galadriel used hers openly in front of Frodo and let him observe the ring upon her finger. Gandalf never did.
However, from the way the narrative states it in the end of the story when Frodo sees Gandalf at the docks is suggestive. It says, "now wearing Narya, the ring of fire, openly", which could imply that Frodo knew of it, but did not say anything due to Aragorn's warning that the location of the Rings is to be kept secret.

The secret flame is the flame of Iluvatar that resides at the heart of Arda. The Flame of Anor is indeed Narya. Nor/Nar refers to the sun (Anar/Anar is the name of the sun, and Narsil meant sun-moon). Gandalf is essentially saying he wields the fire of the sun, which makes no sense unless that is the name of his ring.

It wasn't Gil-galad's choice, nor was Nenya the weakest. All we know is that Vilya was the mightiest of the three.
UT says that when Celibrimbor learned of Sauron's treachery, he gave Nenya to Galadriel (whom he loved, in some versions), saying that he hoped it would abate her longing for he sea (which all the banished Elves suffer from). He gave both Vilya and Narya to Giul-galad. Before going off to fight Sauron, Gil-galad gave Vilya to Elrond (who was, essentially, his heir) and Narya to Cirdan.

While all these reasons are true to an extent, the last is the most correct. It is also the reason why some Elves hoped that when the One was destroyed, maybe the power of the Three would endure.

The Rings did indeed lose their power gradually. That's been stated multiple times.