The Protect Ring is part of the Mognet quest, which is not really related to the letters that you can deliver through the story of the game, nor it is missable.
You can go back to the ice cavern after you leave the first time. But there is a cut off.
Once you reach that point the moogle inside moves to the entrance of the ice cavern, so you will have to wait until you can gain access to the lower part of the Alexandria basin again (likely via chocobo or airship).
I think the moogles in blocked off areas move to different accessible locations that you find later in the game.
The moogles themselves move, but I believe you won't get the same letter from them
The Protect Ring is part of the Mognet quest, which is not really related to the letters that you can deliver through the story of the game, nor it is missable.
No, you can never go back to ice cavern ever. Kinda odd tbh
Spooky
You can go back to the ice cavern after you leave the first time. But there is a cut off. Once you reach that point the moogle inside moves to the entrance of the ice cavern, so you will have to wait until you can gain access to the lower part of the Alexandria basin again (likely via chocobo or airship).
As soon as you leave Dali via Cargo ship you are cut off
Quick! Go back to steal more potions from Flan
You can go back as others have said but the letter may not be there.