I think which Eidolon you have to fight is random. For example, one instance I had to fight that angel Eidolon, another instance I fought Sigrun. So yes, random. How random, as in which Eidolons are excluded based on chest? Heh... we'll need more time and people to find out.
My guess is as many Eidolons as there are confirmed for release. So, for example, that Artemis Eidolon? I do not know if Artemis is even in the game yet.
Also, if you defeat the Eidolon, there is a chance that they will drop their key fragments. For example, I just finished Sigrun, and I got 2 Sigrun key fragments. I received none from the angel Eidolon, however.
Oh yeah, one last thing. That one time per day thing? It's strawberry flavored cake. If you move your cursor to highlight the dungeon name, it will actually tell you that it resets every 360 minutes. Or, 6 hours. It seems to reset when either 6 hours after your last entry has finished cooldown, or if 6:00 on the server time hits, whichever comes first.