There's six different teams that you can set up (Edit -> top-right "Switch Party"), and the primary Eidolon from each one will show up to your Friends. Up to one per element, so if you set Diabolos in team A and Medusa in team B, your friends will only see Diabolos.Nope! There is a stat which helps you do more damage with skills, it's Ability damage. But sadly, that stat is really, really bad and you should avoid it as much as possible.
Well, if you want to know, super-simplified the damage formula is CharacterAttack*ElementalAttack. CharacterAttack also comes from your equipped weapons, but the only source of ElementalAttack is either elemental advantage (like attacking Fire mobs with Water) or Eidolons which have just "Atk" in their description.
As such, the just "Atk" Eidolons are far, far stronger for you than Character Attack ones. Once you get strong, that is. But until then, you can use either and it's pretty much the same stuff.