Does the currently available main story stop at #16 Meeting the Zestians?
AFAIK, no new content was added as of yet. So you can stop dreaming of getting a tit-job from Adele for now.

Do Gems have any use outside the gach, and as an expense when limit breaking stuff?
Yes, you need gems for lb. More stars requires more gems. Other uses are for Union donations to increase union rank.

Is there any way to grind out levels on souls/kamihime quickly? Kinda like the "Maruppe March" in xoverd.
For beginners, 7-4 is a good map.

Are "Idea Core" items used to limit break, or do they have another use? Gotten a few of them from doing raids.
Yes.

I'm guessing from the raids I've done, that they're the best way to obtain the items for limit breaking?
Other materials can be acquired from daily SP quests for different elements.

On some of my Eidolons, in the bottom right corner of their boxes in orange text it has +1, +2, etc. What exactly is that?
Those increase your stats. Each + increases your att power by 3 and hp by 1.