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The Avatar can surely live in a spherical ice and that's what Aang did when he ran away from his home and entered the Avatar State which encaged him and Appa in spherical ice.
But I don't know how exactly Appa survived. Appa could have probably died of starvation or any other reasons. How exactly did Appa survive that long?
Aang didn't survive 100 years under the sea because he was the Avatar, but rather because he had frozen himself and Appa inside a hollow iceberg.
There aren't too many details on how this was possible, but it's assumed that both Aang and Appa were hibernating during the time they were frozen inside the iceberg.
Cryostasis is a feature more common in sci-fi works than in fantasy productions, but we see plenty of examples of sci-fi and steampunk tropes on the world of Avatar that we can accept cryostasis as a thing that could work, especially when there's magic (which in this world goes by the name of bending) involved.
We can see in the episode The Storm (Book One, episode 12) that after Aang found out that he was the Avatar, he flew away with Appa to the south pole and they were caught in a storm, upon falling into the ocean he faints, sinks a little bit, then Aang enters the Avatar State, creates a type of air bubble around him and Appa, and then he freezes the water around this bubble.
It's never clearly stated what was inside the ice bubble that kept them alive, but the same factor that allowed Aang to live, also allowed Appa to live, and this is not related to Aang being the Avatar. Of course, by being The Avatar, Aang was capable of creating the environment that allowed him and Appa to live, but the reason why he was capable to live in such environment doesn't seem to be related with him being the Avatar, otherwise, Appa would have died.
The only reference I could find stating that he was hibernating is an interview given by Bryan Konietzko to the Wall Street Jornal. Highlighting is mine.
Interviewer: If there’s a new Avatar, that means Aang has passed on. If he died around age 70, isn’t that pretty young for an Avatar?
Konietzko: You gotta keep in mind that he was frozen in a state of suspended animation for 100 years, so he kind of burned up some of his extra Avatar time.
Also, as you can see on Konietzko quote, Aang had a shorter life due to spending 100 years frozen on the iceberg. It looks like that he needed to spend energy to keep the environment that allowed him to live inside the iceberg.
