Phantasmal Killer has been errata'd
Phantasmal Killer (p. 265). The frightened target makes a save at the end of its turns, not the start
So the complete exact text is:
You tap into the nightmares of a creature you can see within range and create an illusory manifestation of its deepest fears, visible only to that creature. The target must make a Wisdom saving throw. On a failed save, the target becomes frightened for the duration. At the
start end of each of the target’s turns before the spell ends, the target must succeed on a Wisdom saving throw or take 4d10 psychic damage. On a successful save, the spell ends.
IMHO DM Ruling is required here.
That successful save refers to any save for that spell or just one of the "avoid the damage" saves?
Since the errata says "the frightened target makes a save" I'd also rule that if the target has the frightened condition removed the entire spell ends.
Please note that this strict ruling makes this 4th level spell sub-part.
Feel free to give Phantasmal Killer some DM love :D
After more general considerations and a lenghty discussion with friends I feel the need add this:
Lets consider Vicious Mockery
You unleash a string of insults laced with subtle enchantments at a creature you can see within range. If the target can hear you (though it need not understand you), it must succeed on a Wisdom saving throw or take 1d4 psychic damage and have disadvantage on the next attack roll it makes before the end of its next turn.
If cast on a creature immune to psychic damage should I roll the save?
Yes, because a failed saving throw determines two outcomes, psychic damage and disadvantage on the next attack roll. If the target fails the save it would take no damage but still suffer the disadvantage.
So, back to Phantasmal Killer:
"On a successful save, the spell ends." seems pretty clear to me.
If the save agains fear succeeds the spell ends, if it fails the spell goes on even if the target is immune to the frightened condition. This is brutally RAW.
I very much doubt this is RAI, especially considereing the wording in the errata. In any case, forgoing the first save its a bit too much.
I'll deal with the simplest question first - Horrifying Visage is listed under the Ghost's Actions section, so it requires an action to use. While under the effects of Time Stop, this is impossible for them.
As for the question of multiple Ghosts using Horrifying Visage, the answer is contained in the rules quote you included:
If a target's saving throw is successful or the effect ends for it, the target is immune to this ghost's Horrifying Visage for the next 24 hours.
Immunity is only to a specific Ghost's Horrifying Visage, not to any Ghost's Horrifying Visage.
The immolated creature makes a save as normal and, on a fail, suffers the illumination effect but not the fire damage.
The interesting thing in this situation is that the creature is making a dex save to avoid the immolation. Essentially, they are jumping out of the way of the incoming flames. How dexterous they are has no impact on their immunity to fire.
The rules for making saves (PHB 179) states:
The creature is immune to damage from the fire but consider this: if the spell only made the creature cast illumination, might that, in and of itself, be a "risk of harm" (and therefore warrant a saving throw despite the fact that the target is immune to fire damage)?
It would seem the answer is "yes" because the light cantrip requires an unwilling target to make a dex save. By way of example of how light could be detrimental, consider the Gloomstalker ranger's Umbral Sight ability:
So, I'd argue, the creature still needs to make the save against immolation. But what, then, of the illumination effect? If the creature is immune to fire damage, is it immune to the illumination effect as well?
I can easily imagine a situation where a fireproof object is engulfed in flames (one might say "burning") but, because it is fireproof, it is not actually suffering any harm from the blaze.
In the case of an immolated target, it requires only a small amount of imagination to envisage a creature that is alight but not harmed by immolation's flames in the same way. (Think Khaleesi emerging from the burning Dothraki hut in Game of Thrones).
I think the wording of this spell is a little unfortunate because one might read it as the target is doing the illuminating. It's more true to reality (and arguably intended) that the phrase "the burning target" considers the target and the flames around it as a singular combination (that is, the flames and the creature they are engulfing).
So if the target is unable to avoid immolation's flames, it still does continue to shed light as a result of the harmless flames that engulf it even though it is immune to the damage those flames would normally cause.