OP? Heck spoilers in whole thread!n But wait! They’re confusing, complicated, highly speculative spoilers! So no harm.
First of all, Speaker: yeah, that farewell kiss in EoE was touching in Eva’s usually disturbed way. Or else she was at the end of her rope as to WTH can she do to make him snap out of it 
On the 2 mothers: You have it. Yui (Shinji’s mom) was fully and irreversibly absorbed by 01 (like almost happened to Shinji himself). Kyoko (Asuka’s mom) had part of her mind/soul/chi ripped into 02, what was left behind was a deranged shell of herself. Having permanently absorbed the life-force of a human apparently enables 01 to go into self-animated “berserker” mode – and in EoE, briefly 02 starts to go “berserker” but too late. (BTW one realization that hits Asuka in the end, upon contacting Kyoko’s spirit, is that it is the AT within her that lets her exist.)
00 was crewed by Rei (means both “zero” and “soul”) so all bets were off on that – Rei and Kaworu under the right conditions can generate external ATs and animate EVAs unaided. Which is why dummy-plugs duplicating their imprints could be used to control EVAs w/o need for a pilot, or even over his will.
As to the the suppossedly “special children”, there are hints dropped that MARDUK, the project that ostensibly identifies the special children is nothing but a shell set up to misdirect attention: that NERV simply, crudely and coldly placed in a convenient class at their disposal the children of people who had been expended in the project, knowing that the broken spiritual bonds could be later used as tools.
As to Adam/Lillith/Angels:
Apparently, SEELE’s prophecy for achieving HI required the capture and harnessing of Adam and Lillith, who in that universe we’re to assume were under the Antarctic ice cap for whatever reason. In doing so Adam was awakened, and his capture caused 2nd Impact, after which they apparently saved an Adam barely regenerating in an embryonic state. It seems that in order to achieve HI, they needed a controlled version of the Adam being, so from that Adam sample they then cloned the EVAs (Thus the mythology is satisfied: Adam and Lillith, equals, are separated, and Eva, subordinate, is created from the flesh of Adam).
It seems 3d Impact could be achieved in a variety of ways, and that NERV and SEELE, each for its reasons, had several back-up plans as to how to achieve it on their terms (e.g. Gendo wanted it to happen thru an Adam-Lillith union that he would control thru Rei so as to guarantee a conscious reunion of himself with Yui). However they all relied on Adam, or a version thereof such as the self-animated EVAs, being under their controlling power.
We’re to infer that SEELE’s scroll foretold that the Angels were coming to join Adam, and that if they did so before everything was ready, the eschaton would happen in the Angels’ terms, the humans would NOT be instrumental, get no part in the shaping of the new existence and maybe even just lights out. Thus NERV gets to interfere with the Angels. And creates an eleborate fiction that it is Adam down there (misdirecting both the Angels and SEELE); and any Angel entering there would be a fate worse than death (motivation for the people who were not to know the real truth to stand and fight).
It’s never really spelled out but it is made to look as if there were two schools of why HI was desired: one apparently favored by SEELE, as an opportunity to just ditch humanity and move on to a purified new plane as a brand-new life-form; and one apparently favored by Kouzou and the Ikaris in which all that was good and positive about all those who lived would be preserved and carried on into the new existence.