In the APG it is stated that the creatures summoned by the summoner are summoned from their respective home plane.
So is there a plane that those creatures rove somewhere out there?
If so can anyone find these creatures alone,fight with them,befriend them and ultimately gate them in?
The variety of creatures one can accomplish is wide.
Furthermore,in a campaign allowing the option of monstrous characters could one play a character that is an unsummoned eidolon?
The whole task of forming them rule-wise is completely detailed already in the APG (only lacking the CR but that is trivial as it can be figured out the same way an advanced monster's CR can be figured out) and the logic with which things work falls in place.
Is it rule-wise valid to call a summoner's eidolon with gate (or any eidolon for that matter summoned or unsummoned)?
I don't find anything stopping you,it is an outsider after all and gate calls extraplanar creatures.If it can be dismissaled it can be summoned.
As written on the rules it can be subject to the dismissal spell in turn which sends it to its own plane of existence not out of existence.
As there are more than one summoners there should be at least as many eidolons that are and are not in that particular plane of existence as their summoners summon them and other people or the same summoners dismiss them.
Furthermore if a summoner dies it doesn't say it is the end of the eidolon as well.
So one should at least be able to gate them in.Aside from interacting with them.