It's been a while since I've seen Lain all the way through, but the explanation of "fake" Lain might explain some other things you're confused about.
As far as I understand, the reason the Knights created fake Lain was because of Lain's purpose, to merge the Wired and the real world. By creating a Lain in their own image, it seems that they wanted to alter the relationship between the two in accordance with their ideals/beliefs/etc, though those are still unclear or up to interpretation. This is part of the reason Lain and Masami have a continual back and forth about what defines god, and why he believes that it's people's belief in god that brings its existence. As for the Kids project, the one detail I remember from that specific layer is that it's based on the idea of the collective unconscious, and that the Wired is related to it, whether it be that The Wired is itself the collective unconscious, or that it's the way that we as humans can access it (which is probably why it's named The Wired in the first place). The idea is hinted at when Lain tells the Cyberia gunman that everyone is connected, though that has a much wider purpose beyond its literal connotation.
That's probably all the help I can be at the moment. I've been meaning to rewatch the series since the anniversary.