It is all a matter of timing as to if it applies or not AND if it is a skin layer issue or a whole body issue.
* Outer Skin and Muscle issue: The eidolon becomes a fully covering body suit. It engulfs any current limbs/head/wings/claws/natural armor that you currently have and replaces them with the eidolon's version of those items as well as STR, DEX and CON. If you are serpentine, and have not paid evolution points for arms, you don't have arms and casting somatic spells is impossible without the still spell feat.
Resistances I have seen as being a "whole body" coverage and not a skin and muscle coverage. So, they continue acting on the melded creature. Any ability that is a "whole body" issue is useable in both forms. Barbarian speed increase, resistances, channel energy affinity, etc.
a) Things that are 24x7x365 continuous are applied to the summoner. This is why the inherent ability stat bumps at lvl/4 should be placed on only CON or mental stats.
b) Then things that are the eidolon "living armor" suit are applied, like natural armor and extra limbs, etc. This replaces STR, CON, DEX, Armor, Natural Armor, BAB, limbs, and locomotion.
c) Then duration spells are applied. Barkskin, Overland Flight, etc.
d) Then per round natural abilities. Draconic Sorcerer Claws, Draconic Sorcerer Wings, etc.
So:
* No, Dragon Disciple Ability boosts to strength only applies to the summoner, not the eidolon.
* No, Abyssal Sorcerer Strength of the Abyss only applies to the summoner, not the eidolon.
Yes, the use of Eldritch Heritage to get Draconic Sorcerer Claws is valid because the claws grow from the current skin.
Yes, the use of Eldritch Heritage to get Draconic Resistance to an energy type is valid.
* No, the use of Eldritch Heritage to get the +1 Natural Armor from the Draconic Resistance is covered up and negated while in fused form.
Yes, the use of Eldritch Heritage to get Draconic Wings for rounds per day is valid because the wings grow from the current fused body.
In my case, the mermaid loses access to her +2 Natural Armor of the mermaid, swim speed and breathing water when she summons her suit. She spends a point on Gills so the fusion can breath water. An aasimar synthesist keeps his resist 5 acid/cold/electricity. (Cold chills to the bone, not just the surface skin.)
* A stryx synthesist would have to pay for a set of wings with evolution points, or have his wrapped up and bundled under the eidolon skin, making them useless.
* = See
Synthesist where most of the FAQ/forum questions have been answered by SKR, which I take as authoritative.
SK, aka me, just the local board authority since I have two synthesists in play.
~~~~~~
Tieflings: Fiendish Vessel (Archetype) is banned because it is too evil to deal with. Tieflings don't get the outsider type weapon proficiencies of simple or martial automatically. It has to rely on the classes chosen to provide weapon proficiencies. Other than that, I believe all the bits in the ARG are available for a tiefling character to use.