I had to break the "being type" into two branches: Children and "Lost Children" for the sake of expanding the story/character-generating potential of the concept. Also helps to generate a massive number of filler/extra characters. The former get re-homed, and their fates are left to the True Gods and Goddesses(these either can't survive in the City, or can still be handled and raised by a True God or Goddess with a good amount of risk. Usually both.) So basically the're adopted, where they also get to make their own choices. In a nutshell, its down to "potential." There are a few characteristics that make up such an individual, and there are also common factors:
For Lost Children generally its:
-A very loosely defined origin 'Verse with few defined rules(often associated with Young Gods) OR
-A Godless Creation(very rare, usually spawning very powerful individuals like Forte and Sora)
-That being's "ism" has either outlived its own Creation or Creator, or couldn't be contained by them by the usual methods.
-Enough innate potential to defy the Omnipotence Paradox(i.e. can God create a rock so heavy he can't lift it)
-And the most important thing, consciousness. They need to be able to mentally process The City and the Void Lands, else they end up doing what amounts to self-destructing.
-A very hard life, usually connected to these attributes somehow.
They were first called "Lost" because they couldn't be re-homed(as seen Nana's raw consciousness, uncontrolled, would itself be lethal to True Gods and Goddesses on a massive scale.) So at some point, I think it was during Seicross' era, the decision was taken by every inhabitant of the City, and the City itself to start a project to elevate these individuals and bring them in. Gonna need to check the data points as to the history.
For regular children, its:
-Enough innate potential to at least rival their own Creator, but preferably to equal them.
-Origin 'Verse that's better set up(more experienced Creators, or Old Gods/Goddesses looking for replacements before they fade to the Currents, so that their Creations can carry on)
-Consciousness enough to be able to comprehend their Creators at first sight(these are prime adoption material) but, some can be be slowly acclimatized to the Higher Cosmology.
-A strong being or "ism" that can withstand being in the direct presence of their True God or Goddess.
-Usually led a very hard life from these attributes.
If you thought I have re-structured it very much like the professional driver's journey to F1, you would not be mistaken. BUT. I digress.
These regular children, once re-homed, are usually much more sociable than their new parent. And in some cases, are raised to outperform them. So we do have a few who, instead of spending time at the Halfway House, are given a sponsorship(i.e. kept in the City by someone else's power) to attend either HBG or finish a live-in apprenticeship just like the Lost Children do. These ones, they go home but still keep their ties of friendship to the City and its citizens(this normally does not end well, as they get flooded with visiting Aspects from time to time.)
That's the criteria done. Now for the rest:
I think what you're referring to here is just Nana, who is quite a unique case in terms of blunt force trauma. Not every Lost Child has that kind of punch; in fact she's supposed to be a very rare kind of "tanky" being who's way beyond the status quo(Arashi couldn't move Grey through his techniques when she first appeared, Nana moved her by copying Yuuri's hardest hit and putting everything she had at the time behind it)
For a normal Lost Child, the "output surge" comes from the lack of having a True God or Goddess; since they were built out of the will of their Creator or Creatrix, there could still be some measure of control(however difficult) BUT, if removed from that sphere of influence or if no longer under any influence, well, its what happens when a flywheel goes walkabout in the dyno cell: if their "ism" and consciousness can withstand it, they break out to become their true selves. Most Lost Children are not even in the ballpark of a NODE unit at first, while only some Lost Children have the potential to actually exceed a NODE unit in therms of sheer power output; for example Karu, who can withstand Nikuu's attacks for a short time without being "fully grown" yet. Nana is very much the same, having enough damage soak to absorb Kansai's "Blow of Loli Motivation" but packing more damage output(at the time of her Graduation, she could already cut straight through Nudoru's "Void Phase" and take it to a slugfest)
Their power and potential is completely their own, the only job of the NODE dispatched to get the Lost Child is to either prevent or minimize damage to the True Gods or Goddesses, and their Creations(failure means countless lives lost, obviously) from the sudden awakening of a Transcendent Being.
The City connection, I'll need to wait to get back to Oz and go through the notes to give you an answer for this one. I think its on another USB stick than the Lost Children file(YES, I _AM_ going to get this thing is own dedicated standalone HDD) Sorry about that mate.
^^^Now, all the above(except the part for me being disorganized, which is my fault completely) development came from the first time you asked me those questions. So FFR I do need to credit you accordingly. When I get to the site construction proper I'll need to get your details for the "Developmental Critique" page if you don't mind.