It's important for you and your co-healer to know your tasks. During each phase, you need to delegate who will be healing the tank, offtank, and party. WHM is obviously superior for post-Fireball heals, so if you're paired with a SCH, you should designate specific people to heal.
If you coordinate well, your co-healer can effectively solo-heal during the Fireball/Conflag phase. You should be able to step in to Cleric Stance for 15-20 seconds after every Death Sentence (cast Lustrate, cure once. Then step out of Cleric Stance to pre-shield and overheal before the next Death Sentence). That's more than enough time for a full complement of DoTs; it's really in your interest to make this phase as short as possible due to the MP crunch on the healers and risk of death.
You should average 1.5 Lustrates per Death Sentence. It's unaffected by the Infirmity healing debuff, so it's a lifeline for surviving those post-Death Sentence attacks and Plummets. Sacred Soil is fairly MP-inefficient, but is tremendously useful for preventing bad luck due to huge one-shot Death Sentences. It's best to communicate with voice with your tank on when they are about to eat a Death Sentence without a damage-mitigating cooldown active.
I wouldn't recommend using Sacred Soil for Fireballs; if your party needs to take 3-stack Fireballs, you probably don't have the DPS needed to clear the fight anyway. It can be difficult to maintain Stoneskin on the party and have the necessary reaction time to Adloquium the party member marked for Fireball in time. (The tank and fairy will survive 3-stack Fireballs unbuffed, generally) My advice: stick with 4-stack Fireballs and put less pressure on the healers.
FFXIV-APP is highly recommended. Alternatively, you can ask your tank to keep time manually or use a macro with an internal timer. You will need at least 5 seconds warning to overheal the tank and put up Adloquium.
Eos' Fey Illumination is definitely useful for the final phase Death Sentences, which deal considerably more damage. Due to the timing, though, you'll probably only get to use it once, so I'm unsure if it's worth it. Whispering Dawn can also stand in for heals following Liquid Hell. Selene is definitely better during the first half, and you don't want to use a Swiftcast to re-summon Eos, so I don't think her benefits are worth the cast time.