Stun, Daze and similar stuff don't impede your movement...
...they make your mind funny.
While you are stunned, there's nothing holding your body to move. You just can't think right to actually do something. If you take a really hard blow to your head, that won't make your body harder to move - it will make your brain go gonzo for a few secs, before you became aware of what's really happening.
Stun is not about movement, is about senses of what's going on.
Freedom of Movement makes "your body work right", not anything else. It allows you to move, but not allows you to think. You can't think if you're dead. Or Stunned.
Paralysis don't block purely mental actions, so it don't block "thinking".
Slow makes your body... well, slower, but it doesn't affect your thinking.
Web is... well, a web. It hinders your body, not your mind.
Stun, Daze, Dazzle and similar stuff, on the other hand, makes your senses go wacko, so they aren't really blocking your movement. Stun never stopped you from moving, it just stopped you from thinking for a while - and since you don't think, you don't act.
So, the point is,
If something affect your senses, Freedom of Movement can't help you.
Think like a "Houdini Effect". Houdini can escape from almost anything, considering that he
knows what's going on. If you throw him with a concussion inside a closed coffin... well... he will stay there.
So, to determine what Freedom of Movement removes or not, use a simple rule:
Why I can't move?
If you can't move because a spell or something is hindering your body to move, like Web, Freedom of Movement can help you.
If you can't move because a spell or something is making your brain go gonzo, like Stun, or because your body becomes something that can't normally move, like stone from Flesh to Stone*, it won't help you.
*Flesh to Stone don't impede your movement, it merely limits you to the movement that a stone statue is allowed to do. A "Freedom of Movement"-ed and "Web"-ed person would become a completely untangled stone statue.
Also, read Freedom, the 9th level spell:
The subject is freed from spells and effects that restrict its movement, including binding, entangle, grappling, imprisonment, maze, paralysis, petrification, pinning, sleep, slow, stunning, temporal stasis, and web. To free a creature from imprisonment or maze, you must know its name and background, and you must cast this spell at the spot where it was entombed or banished into the maze.
Emphasis mine.
Freedom removes a bunch of effects, like Flesh to Stone and Stun. It would seem rather... strange to say that a way lower-level spell can do almost all the things that a 9th level spell can. Freedom is Freedom, not Freedom of Movement.
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:
A saving throw—also called a save—represents an attempt to resist a spell, a trap, a poison, a disease, or a similar threat. You don’t normally decide to make a saving throw; you are forced to make one because your character or monster is at risk of harm.
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:
While in darkness, you are invisible to any creature that relies on darkvision to see you in that darkness.
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.
Best Answer
To my understanding, reducing an ability score to 0 causes a condition that can't be avoided by natural resistances called "helpless".
All of this is mentioned in the basics section of the Hypertext d20 srd:
If you're immune to paralysis you can't be affected by things like ghoul touch, but things like hold person affect you (since hold person is a mind-affecting compulsion). Immunity to paralysis prevents you from being physically immobilized through spell effects.
The golems in your example are immune to hold person because it's a mind affecting effect and they have no intelligence score. The unicorn has an immunity to compulsion effects (which hold person falls under as well), which further distinguishes the difference between paralysis and hold.
Loss of ability score causes gain of helpless so even if you were immune to paralysis, you'd still be rendered immobilized when reduced to 0 dex regardless.