Freedom of Movement explicitly disallows the application of two conditions to the subject of the spell, namely they can't be paralyzed nor restrained.
Stunned is also a condition. It is not listed under FoM as disallowed. Stunned also does not reduce the target's speed. It states that the target can't move, whereas other conditions, as you noted yourself, explicitly state an altered speed value.
The paralyzed condition is almost the same as the stunned condition, the only differences being that 1. a paralyzed creature can't even speak falteringly, 2. attacks targeting a paralyzed creature and coming from an attacker within 5 feet of the paralyzed creature are critical hits. Paralyzed inflicts another condition, incapacitated on the target. So does stunned. Stunned does not inflict paralyzed on the target. FoM explicitly disallows paralyzed, but it does not mention stunned, ergo FoM does not protect against stunned, nor against spells that apply stunned, such as Power Word Stun. Considering there's no Power Word Paralyze and no Power Word Restrain, this seems to be a conscious design choice.
I'm not sure I like this, but this is how I read RAW: FoM does not allow movement when its target is stunned regardless what applied said condition to its target (and no matter that stunned is a weaker version of paralyzed.)
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RAI I think the reasoning behind this is that stunned affects your mind, not your speed. Your speed is the distance the creature can cover. When stunned, its speed is the same, only the will to cover the distance unaffected speed would allow it to cover is impacted. (Also, the word "move" in the definition of stunned covers not just the movement of the legs or similar which allows the creature to go some distance. A stunned creature doesn't gesticulate, etc either. At least in my interpretation.)
(PHB 5e 1st printing, 2014, no errata, pages 244, 291-292)
No
A hit is either critical or not; there are no rules providing for "degrees" of criticality. The normal condition for critical hits is rolling a 20 on the d20 without modifiers on one's attack roll - a "natural twenty".
The paralysis effect simply extends that condition. Now any attack (with a lowercase-a) that hits is resolved as a critical hit, natural 20 or otherwise, so dice are rolled twice for damage.
Best Answer
The answer is on page 246 of the DMG.
So in the case of attacking object AC means something different than what it means when it comes to attacking a NPC or monster. So the paralyze condition would not logically apply as an object AC has nothing to do whether it is still or not.