It's not the hardest thing to obtain in Darkmoon Faire, especially following today's hotfix which allows you to see your own rings rather than shared, so I gave it a go and got it done right away. There seem to be around 65 rings, ish. I say 'ish' because it looked like there were something like 10 rings in sight when I'd finished, and a few more I'd missed further out. I didn't pick any kind of route, but I did try to leave lines of rings in case I found myself coming back that way, which I did. But there were no problems. It's very easy, really.
The Blazing Wings have a 1 hour cooldown, but they also last for an hour, so you could wear them all the time. They don't persist through death, but they are added to the toy box for all characters to use.