Dragonvale – Do Limited Dragons Count as Rare Dragons for the Epic Breeding Island?

Yes, the breeding time is always the same as hatching time for all dragons, rare dragons included. This means you can check your breeding time to know what dragon you're going to get; you're getting a Tree Dragon in this case.

You can verify the hatch/breeding time for any dragon with the Wiki's Dragon Reference Chart to check which kind of dragon you might be expecting. Alternately you can just look in the store in-game. Look for dragons matching some/all of the elements of the Parents (combined) that have the same hatch time as your remaining Breeding time.

Note that the upgraded breeding cave/island reduce your breeding time by 20%, but the incubation time will remain the same.

Limited dragons are most certainly not included, I got the achievement without having the Bone dragon, and I also didn't have a Plant dragon at the time either, so I think it only tracks whether you've ever had the dragons, despite the description.

There's lots of speculation on the Wiki and it seems like some people just aren't getting the achievement, but it might be due to game center being laggy or something.

It appears to be the case that you need all non-limited dragons, including rare dragons like the Rainbow Dragon. This doesn't include limited dragons that are currently available; only year-round dragons count, but you need all that are currently available. It seems this is a moving goalpost, meaning as new year-round dragons are added, those are added to the requirement.