[RPG] Can a Synthesist use the Skilled evolution while fused

I haven't personally researched it, but if your rules quote is accurate, then it's perfectly clear: you use the summoner's ability scores, only giving them a bonus if the eidolon has the Increased Ability Score evolution.

A barely-topical rant regarding the synthesist and DnD 3e wild shape:

Synthesist was never an amazing archetype to begin with, because the minor benefits of being able to melee yourself instead of telling your eidolon to do it were mitigated by the loss of action economy - your eidolon could melee just as well without you while you cast buffs and crowd control. The things that made wild shape so powerful, back when it used to replace your ability scores, didn't apply to the synthesist (the druid remained a tier 1 caster, had dozens or hundreds of choices always ready to pick what to transform into rather than one eidolon that can only be slightly modified at level up or by eating a prepared spell out of combat, wild armor, some of the best buffs in the game), so they weren't really as good as a normal summoner, they just had the option to look totally amazing in a way no other character could. The new one is even worse and can only look like specific existing creatures in the Bestiaries. Pass.

From the FAQ (i.e. official errata):

As per the rules on size changes, size changes do not stack, so if you have multiple size changing effects (for instance an effect that increases your size by one step and another that increases your size by two steps), only the largest applies. The same is true of effective size increases (which includes "deal damage as if they were one size category larger than they actually are," "your damage die type increases by one step," and similar language). They don't stack with each other, just take the biggest one. However, you can have one of each and they do work together (for example, enlarge person increasing your actual size to Large and a bashing shield increasing your shield's effective size by two steps, for a total of 2d6 damage).

So Improved Natural Attack feat, Improved Damage evolution, and Impact amulet don't stack with each other. Large evolution and Enlarge Person don't stack with each other. But you can take one from the first category and one from the second.

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Now, despite what the rule actually says, it's possible that it wasn't intended to work that way. Prior to this FAQ, the rule was "Multiple magical effects that increase size do not stack." (quoted from several size-affecting spells.) Under that old rule all of your mentioned abilities would stack with each other because most of them are non-magical. The FAQ both added the clause about effective-size-increases (which is clearly the main point of the FAQ), and removed the "magical" qualifier. I'm not sure whether the latter change was intentional, or just sloppy wording from the FAQ author.