If you don't (re)read them beforehand, I'm afraid you are either going to not understand anything or misunderstand everything.
Introduction:
This is without a doubt the most dense post of all, and because of that I beg you to leave any questions in the comments for I shall try to answer them.
Terms like "anima" (psychologically feminine) and "animus" (psychologically masculine) will be cited often, and to simplify they respectively refer to emotional and intellectual aspects of the collective unconscious (refer to my first post if you forgot what that is — and no, it's not as intuitive as the term seems to suggest).
Thus, on a personal unconcious level, we shall postulate that the former leans towards the Imaginary like the latter towards the Symbolic (and please reread the first post along with its comments in case you also forgot what they are).
The Grand Mother archetype is just a symbolic subrepresentation of the anima, essentially, and entails exactly what it supposes to do, Its equivalent would be the Wizard, who he won't need mention here.
Analysis:
Shinji:
Some background one needs to recapitulate on Shinji is that he literally lost his mother early; likewise his father, metaphorically, which are the main figures to help developing an individual's (conscious) psyche to adult(s') standards — the former specially with aspetcs of the anima, or feminine side (or the Imaginary, as it shall be attempted to draw a bold pararell), and the latter with those of the animus, or masculine side (or the Symbolic).
Jean Piaget teaches us throughout his entire bibliography (particularly "The Language and Thought of the Child") that the child's development of cognitive abilities and consciousness itself highly depends on his exposition to adults, and the necessity to convey one's thoughts to them, therefore being forced to entail rationality, form, structure and procedure to their feelings.
Both insights above are of importance: if Shinji lost the two adults most capacitated and responsible to help him out with such a development, then it becomes clear why his emotional, anima, Imagenary and whatnot; likewise intellectual, animus, Symbolic etc. aspects appear so unconscious and at the same time out of bound(ary) or organization throughout the whole series, thence resembling Adam (for his entailed primitive organization; simbology).
So Shinji, like any kid's merrily programmed for birth and beyond, is in search of a soul (and as I know hardly anyone here will take the reference, it's one to a jungian book), and as already mentioned in my last post, the narrative does no more than represent his path towards such strive, namely of becoming master; Subject (lacanian reference) of his unconscious, as Adam attempts to overtake Lilith (in search for psychic balance).
For SHinji doesn't know how to formulate such demand (or convert it into a [genuine] desire, like Lacan begs), therefore he projects it (and that's basically the dawn of any analytical process)... on everyone — Gendo, Misato, Rei, Asuka etc., whilst likewise introjecting their insecurity (or uncertainty) in dealing with him; in summary his self-imposed crime and thence jail.
When facing Sachiel ("The Price of God"), the 3rd angel of our last post, he pays the first "price" for finally "achieving his soul," namely forsaking it to someone else as every kid does after being born (much further elaboration on my first post and its comments, which I alas cannot hope to summarize well here), namely his father — and with Shamshiel he as expected reaches towards his unconscious anima (since he's a man, his animus should be already conscious, despite heavily depending on Gendo for [its] validation) by protecting his friends and therefore see a reason beyond mere reason to fight, which would be to protect his classmates (see episode 3), an idea further developed thanks to a bond formed with Rei when confronting Ramiel (the 5th angel).
So Gaghiel, Evangelion's Levianthan, said in the Bible to only being prone to extinctiong by God, is defeated uncoincidently along with Asuka's appearence, as if she was something numinous out there in the world which unconspicously appeared to wreak [w]holiness into his mind; as if both had together ascended psychic apotheosis (inflation), but only for a short while until the real descent into unconsciousness begins.
Asuka:
Asuka is much like Shinji, in the sense she lost her mother early (also psychologically, indeed), but her farther, albeit alive, seems to be a somewhat missing figure in her family, though not as badly as Gendo.
This at least should have allowed Asuka to develop to some extent her animus and Symbolic, which in one hand tend to backlash due to the more underdeveloped anima and Imaginary — thence she becomes crushed by the Other('s gaze, as is often the case with the Symbolic), desperate for validation just as for, on a more intimate, feminine note, attention —, while in the other she's highly upfront, feisty and stubborn, yet cold, distant and (frankly, but surprisingly) hollow at the same time.
In that regard, Asuka also hungers; even suffers for a soul, and Shinji's regressed nature is exactly what she'd be looking for (namely her own projected; maybe even idealized unconscious, and eventually [w]holiness), albeit admitting to it fundamentally contradicts how she views herself (the reason why Kaji looks much more appealing to her), thus introjecting an image of her more instinctive socioemotional aspects (as often partaining to that mental modality, firsthand — particularly the anima) becomes out of the question (surely, the unconscious has its reasons to be afraid of becoming conscious).
The difference here is therefore that Asuka projects into others instead of onto them, being in a sense as boundless in consciousness (or Lilith, as you might remember from the second post) as Shinji in its counterpart (Adam). They're like yin and yang to each other, while Rei's higher similarity to Asuka but likewise apparent disregard for their similar problems, makes her seem preposterous in her eyes (aside her threatening proximity to Shinji [and to some extent Gendo] which to an extent makes Asuka feel insecure).
Asuka's first appearence with Gaghiel's defeat is a good introduction for her (psychologically inflationary) character. Moreover (and because of it), she's the perfect target of Shinji's twisted heroic journey's projection (towards individuation, as chronically the case with the Hero archetype, almost alawys represented by the protagonist).
As already foreshadowed in my tihrd post, our next angel's name, Israfel, "means 'The Burning One,' which is interculturally reminescent of 'passion,' but also 'Hell,' pointing to the ups and downs aced by Shinji and Asuka in order to defeat him," whence "The Brother" is called where "Asuka and Shinji show an uncoincidentlly higher degree of cooperation in order to defeat him. Humanity must therefore unite against its numinous psychic matter" (all according to theory, like I promised before).
Such process of cooperation eventually stumbles (as with many cases of unrequinted/toxic relationships in general): Matarael's gaze (like the Other's) menaces to breach their Absolute Terror Fields (A. T. Fields, like a psychic defense mechanism, long ago described by Freud) and constitute a "Premonition of God" (his name, literally) as they can't manage to hide each other's shadow from one an-other/a-no-other throughout the series's development. In fact, Shinji becomes [w]hol[l]y exposed by Leliel (the 12th angel) just like Asuka eventually by Aerel (the 15th angel), and even Rei by Armisael (the 16th angel).
Rei:
As Lilith's vessel, Rei is thenceforth almost completely devoid of development when it comes to her anima (and I know it looks weird to see Lilith as representing the animus, but please don't take symbology too intuitively) and thus most instinctive/primitive (or Adam's) aspects, justifying much of the disregard towards preserving her own life (like the attempt of self-destruction against Zeruel and later sucessful one with Armisael, albeit on a more caring note for Shinji) or species (with socialization and sexuality).
That doesn't mean she's completely lacking in those regards, and interactions with other humans (particurly Shinji) help her develop these basic abilities so lacking (albeit the same in his case, but not so badly) due to Rei's special circumstances, at the end betraying Gendo (specially regarding the manga) and (in this case genuinely) reaching apotheosis, have come to terms with her more feminine; maternal; nourishing side.
Rei, as Yui's clone, symbolically represents how a mother's child (in this case, Shinji) can help her ascend psychologically, or with other words — explore more the anima aspects of her personality, whilst taking time away from the father/Gendo/Symbolic/Lilith, fusing at the end with Adam. In the other hand, she also helps Shinji come to terms with his anima (more specifically the Grand Mother archetype, for his anima itself was still to be faced against Asuka on "One More Final: I Need You"). She's the first to fist induce him to pilot the EVAs and come closer to his dad like any mom usually does (long story).
Gendo:
We don't know as much of Gendo as I would like, but since he's a side character, that's to be expected. It's easily presumable he has an underdeveloped feminine, anima, Imaginary side, which as a consequence remains unconscious and at the same time influences all of his behavior and posture towards life, particularly the overdependence on Yui, a source of emotional support. It's interesting to note he works to NERV like his wife was associated to S.E.E.L.E. ("soul," in German), further revealing the contrast between anima and animus.
It's obvious his influence on Shinji though, especially when it comes to Bardiel, the 13th angel, forcing him to almost (or effectively, in the manga) kill his friend, leading to Shinji's fallout on his Symbolic order, thence nearly losing all motivation to live if not for Kaworu's appearence. His death — along with Asuka's comatose and Rei's suicide — further brings him to a mental breakdown as he becomes ever unable to develop his psychological abilities with the help of others (as in principle any human does).
Doubtlessly, Gendo is also in part important to Rei on Armisael, the 14th angel, when she attempts individuation by self-destruction from him altogether, killing along with herself the Symbolic order which so shackled her. As both Yui's clone and Lilith's vessel, she's the perfect image of his strive to reach the unconscious and become whole (selfishly and alone, unlike S.E.E.L.E., which wanted to unite all of humanity into a single being; not just Gendo and Yui, thus also offering a pararell between collective and personal unconscious).
Yui:
Yui played as much an influence on Shinji when it comes to the twelth angel as Gendo did on the thirteenth, and the fact both are next to each other on the line shouldn't be taken coincidently, as the former represents a(n abrupt) contact with Shinji's anima as the latter with his animus.
She also serves as Shinji's EVA (or Eve), a testament of (his) humanity's existence, constituting so to speak a feminine counterpart of Adam, whom lilin, children of Lilith, much needs in order to achieve [w]holiness. That's why her sacrifice was such an important, vital part of S.E.E.L.E.'s projects.
Yui plays a strong role on Shinji's contact with himself against Leliel, allowing him to come more into contact with his Grand Mother archetype until he finally became psychologically prepared to leave the EVA's "womb" (which he refused to leave after the fight).
Ultimately, Yui's plan seems to have been bridging the possibility of "individuality" across intrumentality (which she was not able to fully prevent within her limitations) for humanity, operating in this sense much like Eve (see: https://wiki.evageeks.org/Theory_and_Analysis:Yui%27s_Agenda).
Kyoko:
Kyoko's imagery is the one to mostly overtake Asuka's mind during the conflict against the 15th and 18th angels. She seems to represent an insane version of Yui due to losing her mind thanks to the contact experiment, leading to Asuka's more damaged anima aspects than Shinji's.
She confuses her with a doll of hers and therefore builds the foundation of Asuka's difficulties in dealing with her emotions. There are also signs of highly unhealthy symbiosis, paving the way towards her need for attention and individuation, as if to recognize her own individuality.
Her suicide can be taken as a step ahead from losing her S.E.E.L.E. ("soul") to EVA (or Eve) and then afterwards likewise her NERV(e) ("wits") due to her husband's affair, becoming devoid of either Imaginary or Symbology, albeit this is already highly presumptive (and difficult to really elaborate).
Kaworu:
Kaworu, conversily to Rei, is Adam's vessel, and nonetheless posesses Lilith's blood, therefore amounting to a being of truly numinous and harmonious constitution. He comes in as a substitute for Asuka due to her unstable mental condition, not only for NERV but also Shinji, who struggled to keep her at arm's length. Unable to come in contact with Lilith yet, for the unprepared world would blow up with such contact, Nagisa also offers him the chance to choose between instrumentality and individuality on The Third Impact (when humanity was ready for it), living up to his real name ("Tabris," or "Angel of Will", similarly to Lacan's concept of desire). As in my other post — "Like Jesus (according to tale) he sacrifices himself for us, despite the pain allowing Shinji and others to finally 'ascend' psychologically, as the plot continues to promote the development of its characters".
Mari:
Her name, in the context of Evangelion, shouldn't be taken lightly: she's Shinji's (holy) Mari, just like Yui is his Eve, Misato Lilith/Eve's substitute, Rei Lilith/Eve's vessel, and then Asuka his Lilith.
In the Rebuild series, Mari is responsible for essentially firsthand providing Shinji light; [w]holiness (just like Mary to Jesus). It kind of drops the issue to be discussed ahead in the conclusion, though, but still.
Conclusion:
Shinji and Asuka side by side, like Adam and Lilith; their vessels watching from afar. Lilin (represented through "Rei's" corpse) died for the ascencion of humanity. I guess it's time my posts soon reach their conclusion, as the snake bites its own tail in mandala, namely by uncoincidently referring you to my first one — "One More Final: I Need You".
Epilogue:
Next I will probably try to organize all of these convoluted thoughts into an affordable (albeit still dense) text for everyone, besides drawing my final remarks, thoughts and conclusions on Neon Genesis Evangelion, what it in the end really means, and why this franchise is so psychologically paradigmatic.
