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.
You have the right of it; each beam gets a boost
Note that the Mantle of Flame says
Whenever you roll fire damage on your turn, the roll gains a bonus to equal to your Charisma modifier.
Compare that to the Draconic Sorcerer's Elemental Affinity which states:
when you Cast a Spell that deals damage of the type associated with your draconic ancestry, you can add your Charisma modifier to one damage roll of that spell.
Spells like Scorching Ray which have multiple attacks will generate multiple damage rolls. Though they are all damage rolls of that spell, the rolls themselves are still separate instances of rolling damage. Since these are independent rolls and all occur on your turn, the Mantle of the Flame bonus would apply to each of them because it lacks a "one damage roll of that spell" clause.
For completeness, the Warlock invocation that you listed also lacks the "one damage roll of that spell" wording, which is why the boost there also applies to each beam of Eldritch Blast. The Evocation Wizard however has the same "one damage roll of that spell" stipulation that the draconic sorcerer has.
Best Answer
Catching on Fire is an Environmental effect; Phoenix Blood Arcana does not apply*
*usually, see below
The rules for Catching on Fire are in the Environmental Rules section and an exception to the normal rules that instantaneous fire spells do not light fires nor deal continuing damage. If a spell references adding that condition (and does not have a duration), then the Catching on Fire is not part of the "damage the spell would normally deal".
There are exceptions; for instance, Touch of Combustion references also dealing 1d4 "damage" (with a Reflex Saving throw) to any creatures adjacent to ones on fire. This is damage the spell does in addition to the environmental effect of being caught on fire. The 1d4 would be healing (while the poor person on fire would still take 1d6 damage from the very real fire).
Alchemist Inferno is not an exception because it is still inflicting the environmental rules for catching something on fire; it just increases the DC because honestly DC 15 is a joke by the time the Magic Trick is available.