Okay, everyone calm down and don't take this seriously, either you will go berserker (like me) or just melt your brain, just try to be comprensive and reasonable, and be open to others interpretations (as butter_pat_head
is). This will be long, so take read with patience, I'll continue since I left:
...Rei was the only human with a soul and the Eva 00 worked with the Dummy Plug, an artifact which emulates a soul, so both the Eva 00 and Rei have identical souls. The other Eva that never has a soul was the Eva 03 (Toji's Eva, the black one), however they (NERV?) also wanted to use the Dummy Plug in it so the pilot would be able to command it.
Going backwards in the history, the case is that God started sending one by one the "Angels" (named an "apostle" in the original Japanese version, anyway it refers to the sons of Adam), the contact between one of his sons and it's father (Adam and and any Angel), would start the 3rd Impact instantaneouly. But what the Angels didn't know if that Adam isn't not in NERV basement, but Lilith is...
For example Kaworu Nawisa (**), goes in search of Adam, he realizes that is Lilith and he ask himself why there is to annihilate all of them. To refresh the memory, Kaworu reachs there thanks to the Angel Lilim (the Production Model: Eva 02 / Asuka's Eva), Lilim is made by humans for combat, so it is the best fitted to represent humanity (said in another way, Lilim is the human race):

Kaworu Nawisa, also know as the Angel 17º named Tabris:
Some Magazines said that he is Jesus, and that his mission is to give choices to humanity, as he is known as "The Angel of Free Will", one of the reasons that he has to not continue fightning is because he fail in his try of find Adam, so he decide to surrender and give pass to humanity, this is arguible (I bet many of you wondered why he almost suicided)
The only Angel that really knows were Adam is and goes for him, is the most powerful of all (surely is), the Angel 14º Zeruel (***), this one entered NERV really easy (compared to the others), and as you may remember it stopped in front of the supercomputer MAGI which control NERV and it's scientists, included Gendou Ikari, and try to kill them...the question is why do it tries to kill them, that's not at all the Angel mission, human annihilation may be a collateral damage...
The reason is simple Gendou ordered Kaji to bring him the Adam's Embryon, Gendou implant it in his hand and so he could unleash the 3rd Impact, so he and his wife (Yui) could become Gods. That means that Zeruel wanted to touch Gendou right hand and unleash the 3rd Impact not controlated by humans.
(***)

After the series finish (Episode 26), Seele sees Gendou's true intentions and decide to take NERV control using the force...attacking MAGI, later NERV with infantry, and finally Evas to destroy Evas (fire to kill fire). Even so, Gendou tries to continue it's scheme of human complemention, which was supposed to be realized in the next way:
Join his hand to Adam to the soul of Rei once there is no ATF, once in unison Rei would become Lilith (as Rei is a clone between Lilith and Yui), and so Lilith would unificate with the Eva 01 and that would unleash the 3rd Impact, however what happened was the next :
Gendou tries to unite to Rei, but when he put his hand inside of her (without an ATF), she takes the entire of Gendou and Adam's Embryon, so Rei now with the Embryon of Adam was able to unite with Lilith and with the Eva 01 which ended becoming the soul of Yui, Gendou's Wife and resident soul of the Eva 01, and so a new god (not physic), Shinji (the son of Gendou and Yui, pilot of the Eva 01) is now the chosen to decide the faith of human race, that means if he will fulfill the "Proyect of Human Complementation" (where the or a human ascend to Gods, I believe) or everything will start over again...So the souls could keep floathin in the void or just return to the Gouf Room, where they would be called again and returned to Earth...
Then the Final comes and is one of the most complex moments, when we see Shinji and Asuka and Rei's phantom (or soul) just floating everywhere...well, most people say that Shinji becomed Adam and Asuka was Eva...this has a solid argument in the chapter Asuka attack NATO (OTAN, in other languages) and SEELE says: "Eva is giving troubles again", but they don't refer to Asuka, they referred to the Evangelion, this is so weird that the Author (Anno Hideaki) needed to make a conference in order to clear this matter, again other point that is not seen in the series...
And so on the End of Evangelion is the start, Shinji as Adam, Asuka as Lilith (yes you heard it), and Rei as Eva...the explanation is easy, Adam and Lilith relationship was very bad, as Shinji and Asuka was, while the opposite Adam and Eva relationship was good, as Shinji and Rei was, as Shinji and Rei were in love, though they never say it honestly, just in case Rei mention it in the EoE....
So Shinji (Adam) needs to kill Asuka (Lilith), in order to give place Rei (Eva) to enter as you remember he grabs her by the neck and goes berserker, but Asuka reaction is so cold that he start to think about and he don't kill her....
Opinnion from the original source:
Until this point everything could be possible, but the logic would be that he kills her in order to be able to let Eva join the paradise with him, and everything start again.
------------------------------------------
My oppinnion:
A - Adam (Shinji) goes searching for Lilith (Asuka). I bet this means that Eva remained alone and was abandonded in the paradise, it won't surprise me if she ended making the 3rd Impact because she was really pissed by Adam
B - It seems that the "logic" that is named above, refers to God's destiny, as it was meant to happen, by the counterpart, if Shinji makes his choice and don't kill Asuka (Lilith), that means he was able to decide by himself (and so he frees from the destiny that god have prepared, or just God's will), and if he wanted to remain with Lilith it would be so, and not as God wanted...
C - He goes searching for Lilith, I bet this mean that he somewhat realizes that he love, or at least she wanted her for a very good reason.
I mean they fight a lot, but that doesn't mean that they weren't in love...., God didn't wanted Lilith, but wath was what Adam wanted ?, after all why he goes searching for her if there wasn't anything special?
-----------------------------------------------------------------
What is really left to understand missing points here:
1 - The Red Cross Book (the ordered seended to Japan, Neo-Tokyo-2
2 - The Longinus Lance (there is no special meant to the series, just what appers in religion), the question is why Lilith has it attached to it's chest...aside from this, this is the true proof that an Angel/Eva and an AT field can be damaged or destroyed by something without neither an ATF or a soul, and even better, something as a Mass produced weapon is...
3 - The bad time that Anno's Hideaki suffered while making NGE (yeah it's necessay specially for NGE)
-----------------------------------------------------------------
Some quotes, though this has been answered already above :
While not AFAIK officially confirmed (we only have what happens in EoE to realy confirm this) , it is popular belief that Rei has the soul of Lillith but she has been made in the image of Shinji's mother, Yui.
She is supposed to be a clon between Lilith and Yui, that's why Gendou was so...well she take care of herself as if they were really close, and as he wasn't close to Lilith of course, Yui was what was left...
While it is common knowledge that in NGE humanity is the spawn of Lillith, some people have suggested that the Angels are the spawn of Adam.
The Angels in NGE are named like the sons of Adam, and in the Japanese version they were referenced as "Apostles", but they meant the same...otherwise the is no real reason to cause an Impact when both enter in physic contact.
Yes it is confirmed, humanity spawn from Lilith the 2nd Angel, humanity so are named the "Lilim", however as Adam is the 1st Angel, and somewhat she is the spwan of Adam as the other Angels are, it's cosidered that the humanity somewhat is also considere partially the spawn of Adam...
Animation is good, humor is good, battels are good, maby it even has some hidden meaning that is good, but let's fact it - the story sucks.
The combination of humor and battles is something which makes it different from any other mech series, if you see Gundam W or most Gundam series there isn't nothing of humor, this is not to blame, most people just see a mech series like Gundam and makes a vague an and partial description as "two robots fighting, end of the series", while Gundam just tries to reflect war among all of it's series, and maybe it does too much, Macross somewhat takes more a romantic point of view for example....
One of the reasons I could never finish to see Escaflowne was specially that it lacked of humor of any means, I mean, well maybe in that alternate universe it didn't exist, but after a few episodes I was ready to enter in hibernation state...I didn't really see the rest of the series, so...my opinnion just apply to this brief start I see.
But yes, I like more combination of situations and emotions, than just fight, fight, fight, war, war, there are some other mech's series which are like NGE and combine episodes and situations which are quite opposite frequentely....