Israfel is a different thing because it was a his body that was split in two, not his soul, and since he could go from one to two and back it's not the same as Kyoko or Lilith/Rei.
I'm pretty sure we don't know what became of his soul when it split. Seems reasonable to me to assume it split just like the body, given that he had two cores and all.
That said, we're still not totally sure that 00 is Lilith, as high as that chance is. And even then, Lilith is a god; it has a lot of soul to go around and probably can't count in the same way that human souls work in the 'verse.
No, Lilith is a Seed, and has a human soul like anyone else. Her connection to the Lilith body might make her special somehow (probably does, IMO), but it's not like there's something divine about her or anything. And don't forget, there are at least three soul bits at any one time -- the bit in Rei, the bit in Unit 00, and whatever remained in Lilith (sufficient to say "welcome home", at least). And of course that's assuming the bit in Rei I was recycled into Unit 00, and that Rei III likewise had a recycled soul. It's possible both of those souls were unique, leading to a five-way split (supported by the fact we see three Reis and a chibi Rei in the ends, but it's anything but a settled matter).
As for Kyoko, there's not much I can defend - her soul was undoubtedly split, but I always thought it was more of a rip or tear that happened in the soul-to-core transfer rather than her purposely doing it.
How is that relevant? Who said the Adams in NME had to split up voluntarily? That is why the theory has so much going for it IMO -- we have a voluntary split (Israfel), a forced split (Lilith/Rei), and an accidental split (Kyoko). To say that one of these might be at work with the Adams is about as uncontroversial as it gets. That's not to say it's a given, of course, or even all that more likely than anything else; for all we know the Adams might be the equivalent of Seeds or something, with something as yet unseen being the First Angel (unless they've been identified as such somewhere, I dunno). But IMO it's certainly a plausible notion.
We did - I guess I got that mixed up a bit. But I still stand by my thought that Adam didn't split.
If it did, where did the pieces of its soul go? And why would there have then been 4 Lances? Do those split as well? One could argue that a piece definitely went into Kawuro, but what of the other 3? Into Units 01, 02, and 06? Or maybe they were transferred to the pilots (barring Rei), which could explain Shinji's red-eyes.
Anything's possible. I think the disposition of the soul bits depends on what exactly happened to the Adams during 2I. I think it's premature to stand by much of anything at this point; like most of our unanswered questions it's a matter of "dunno, wait for Q".