Is there a set rule on how you can obtain your grimoire?
Nope. It's up to you how you get it. Could be from picking it up in a junk pile, inheriting it from your dying friend, having it given to you on your 16th birthday, etc. Think of it as a magical item that binds to potential Espers. The important thing to note is that Espers are naturally drawn to locations where grimoires are: places with highly concentrated mana hijinks.
Does the world in general know about the existence of espers? My character is going Freelancer and I'm wondering how she would learn about the app.
The world in general does not, but those in power are aware. It's more likely that she learn of the app through contact with another Freelancer while she's milling about and trying to figure out how to Esper.
<Snipped quote>
Would having them grow up in Pax Septimus be way out of the question, in this case?
No, that's fine.
Does Bend basically replace a cast note? Or do you still have to target a desired element first via touch, projectile, spawn etc.?
Naw, Bend does not replace the cast note. Imagine it like this...
In Avatar, water benders have that thing where they draw water from a puddle, before then launching it towards an enemy. This would be Projectile + Bend, as the 'vehicle' for the effect of the spell (in this case Damage), is still its projectile form, while the Bend modifier only informs upon how that projectile came to being.
Sorta the reason why most Cast Manipulation notes cost 0.