I understand that Fiora is not targetable by point and click abilities during her ultimate, but what happens during a type of AoE crowd control, like Sona's Crescendo or Amumu's Curse of the Sad Mummy?
Fioras Ultimate gives you basically the same "Invurnerability" as a Vladimir W or a Fizz E. While you are completely untargetable and immune to AOE you will still take damage from DOT abilites. Also something like Fiddle drain will not be interrupted if it was applied before you used your ultimate. The only champion that has a total invurnerability is Shaco because his ultimate makes him disappear from the game for a short amount of time.
You can read about all the different types of Invurnerability here (the one of fiora counts as a "Dash")
No, she can not be affected by anything new while her ultimate is active. I will find out if any effects from CC originating from homing projectiles (Taric, Sion, Vi's ultimate) will still apply afterwards.
But no, during the ultimate, she is not affected by AOE, just as Zed, Master Yi, Shaco during their "disappear" frames.
while she cannot be CC'd in any way during her ult, she can still die (thornmail) while untargetable. and earlier cc that was cast on her(Morgana Soul Shackle(R) to name a few) will resume but will be treated like she didn't ult'd at all. ex. morg R is 1.5 secs, u disappeared for, say 1 secs, u will be stunned 5 secs when ur ult is finished.