Good guesses, but not really how it works.
Both "Character Atk" and "Dark Attack" (what we call "Elemental" around here) buff all Dark units. So you can go and bring any of those three against any element, as long as the team you're bringing them with are Dark Hime and Soul.
The damage formula super-simplified is
baseAtk*CharacterAtk*ElementalAtk
This means that a combination of both Character and Elemental is the best. HOWEVER. Notice that Character Atk is the same as Assault that you gain from your Grid. That means that if you've leveled up your Dark Assault weapons, Elemental Eidolons are vastly superior. That's the short version. Then there's the long version which goes into elemental advantage and Soul weapons and stuff, but that should get you started I hope.
But if you're just starting out (playtime less than a month), the most important thing is: reroll. Reroll into a 100% Elemental Eidolon. Those things are super important for you.