Although I can only speak from the White Mage perspective my partner in crime is a scholar and she rocks, so I'll give some insight on what she does during the fight.
Downtime:
She tends to either throw out heals preemptively, getting Aldo's up on the tank, throwing out Physicks if she's expecting a big hit, etc. But she is also comfortable enough with my healing to throw out dots on certain enemies, such as the Conflags and the big snake when all 4 little snakes are dead.
Big Snake:
With our setup our main tank only takes 1 stack of the debuff, its just how we position them, the 2nd tank however ends up taking three stacks. I typically take on all healing once all 4 snakes are dead, and she will throw dots on the Big Snake and focus on helping getting it down. However if needed she will throw out a Lustrate.
Raid Healing:
Phase 1 we both let the fairy heal people that are hit with fireballs, unless they are really low, we both focus on the tanks through this phase.
Phase 2 is mostly the tanks but she will help with Succors for Fireballs, although I will typically get people back up with Cure 3's and Medica 2's depending, so she can focus Lustrates on the tank when Death Sentences hit.
Phase 3 there's typically only 2 or 3 people getting hit, and in our case it's the Dragoon, Off-Tank and Main tank, she focuses on mostly the Dragoon and Main Tank, and we both periodically throw a heal on the Off-Tank but he doesn't get hit that hard until the 2nd set of snakes.
Phase 4 and 5 is all about communication with the other healer, if you are Stunned, being chased by Fireballs or the Black ball of death (in case your off-tank can't absorb it for whatever reason), your other healer needs to know so they can prepare to heal big hits and keep the tank topped off. As far as Raid healing goes try to position your fairy roughly in the middle to heal anyone running around, but for the most part I'll just toss Regens on people that need it and that pretty much takes care of it.
Fairy:
Honestly haven't paid much attention to this, but I don't think she changes it out, and if she does it's probably like you have described. Sorry I can't be more helpful on this subject, I'm usually to busy healing to pay attention to fairy switches if she does it.