Shinji is attracted to Asuka
The two other main women in Shinji’s life are Rei Ayanami and Misato Katsuragi. He seems to care about them both dearly, but with them it is more difficult to determine the exact nature of his feelings.
Well, what? Shinji is attracted to both Asuka and Rei, which is explicitly portrayed both in the series and in the End of Evangelion "continuity", during instrumentality. Both Asuka and Rei are portrayed as having feelings for him in return as well (and in both cases there is some ambiguity due to Kaji and Gendo respectively). Misato is a bit of a weird case when you consider End of Evangelion (the kiss scene, as well as various scenes during instrumentality complicate things), but has motherly feelings towards him otherwise. So is Shinji's attraction towards Misato which, even if hinted to be that between a child and his guardian, comes off as more ambiguous (and then again Freud).
And as much as the wiki tries to pitch LAS as canon, there is no clear hierarchy between the 3 relationships. There's just a clear hierarchy between two sets of relationships - {Asuka,Rei} and {Misato} - and why is that is pretty clear when Asuka and Rei are each others' foils (the alternatives, two sides of one coin), making them the main point of the few and "subtle" romantic developments from Shinji's life.
He seems to care about them both dearly, but with them it is more difficult to determine the exact nature of his feelings. Shinji appears frightened by Misato’s attraction to him, and though he is kind to Rei, grows to fear her after discovering her connection to his mother
And the glaringly omitted continuation to this is Asuka is in the same boat. Did you forget that he strangles her? Twice? He goes through a cycle of attraction and rejection with all 3 main women in his life (and on a side note, all male friends in his life are taken away by plot). It's part of the narrative. No, it's crucial to the narrative. "Hedgehog dilemma" anyone?
the reasoning is not explicitly stated
The reasoning doesn't need to be stated. It is obvious in all 4 cases (let's not forget Kaworu, Toji and the like): humans are social creatures. That's all that matters as far as the narrative is concerned. And of course, they're teenagers. Eva pretty much spoonfeeds us these aspects several times, doesn't it?
Again, a central thematic point that seems to be ignored for the sake of shipping. Creepy.
It might be that her bullying sometimes causes Shinji to bite back at her, and he views Asuka, in a strange fashion, as a source of strength
Well, fanwank much? Or maybe external material (say, a certain card game) sipping through where it simply does not belong?
one that he misses when Asuka is laid low by her encounter with Arael
Cherry-picking much? This is also the case with Rei in episodes 6 and 23, for example.
Oh, and I didn't even begin to shove my hands into this blunder: http://wiki.evageeks.org/Theory_and_Analysis:Shinji_and_Asuka%27s_Relationship
The complex dynamics of their relationship quickly become the heart of the show
Uh? The heart of the show is (was and always will be, shipping be damned) Shinji's character/person and his relationship with all the characters in the show, not just Asuka. Of course, some (Asuka, Gendo, Misato, Rei and Kaworu) are more important than others.
Most aspects of the relationship, however, remain unspoken or denied and it is important to realize that what the characters say or do almost never directly corresponds to what they actually feel or desire. Thus it is up to the viewer to piece together their motivations from the unspoken or implied aspects of their actions.
I really like this part. I really do. Especially because it applies to every single one of Shinji's relationships, not just his relationship with Asuka. I like it because it's captain obvious material. But obvious things are the best, and they are the kind of thing that belong in an impartial wiki. Not shipping.
In many cases the clues to unraveling a scene can only be found in conjunction with other scenes.
Or, in translation: "in many cases the clues to unraveling a scene can only be found by wearing your shipper glasses."
Are They a Couple? The short answer is yes
Bwahahahahahallshit! BULLSHIT! Really.
but certainly not as traditionally portrayed in this sort of story. [...] Rather, it is the evidence presented in the show makes it clear that they are the show's primary couple.
Oh, how "cute". *rolls eyes*
Now, since that's out of the way *pheewww*:
Friends see a relationship of which the character's themselves are blind.
*crickets, crickets* What? No, I mean, obviously true. But since when did this acquire a romantic spin/nuance and isn't just classmates being classmates making fun of the situation? Shits and giggles.
After Asuka runs from the room Hikari says to Shinji: "Go after her! You made a girl cry!" [...] it's hard to understand exactly how Shinji is responsible. If anyone made Asuka cry it was Misato. [...] Hikari has already figured out that they are a couple, and feels that it is Shinji's responsibility to protect Asuka from this sort of humiliation.
How. What. Why.
IT IS Shinji's fault. And it's obvious. It's both of their fault actually, but the key point here is that Shinji shares the blame for not dancing properly with Asuka. So it's his responsibility to fix this or they'll never get over it.
How come he can find it in him to dance properly with Rei but not with Asuka? That's the issue - he knows how to dance but for obvious reasons doesn't dance well with Asuka. How is Misato to blame that Shinji and Asuka don't dance well together? It's just the opposite, what she does is to give them a push to get over their damn hormones and do the thing like they're supposed to.
the kiss
FINALLY, something with a little more weight than the "right" colours under the shipping spectrometre. Yes indeed, there is sexual tension between Asuka and Shinji (how couldn't there be when they're two teenagers living in the same apartment? it would've been weird if stuff like this DIDN'T happen) - a couple make them it does not. The article momentarily (and "conveniently") ignores and omits the aftermaths of both instances. Shinji does not go in for the kiss, and Shinji does not go after her in episode 15.
Since we're at episode 15 (and because I'm not in the mood to write more in this post), I think it's proper to mention Shinji's dialogue with Rei which is also fuel for shipping curtesy of Anno and company ("you'd make a good wife" - just as standard a trope of "romantic" development in anime as the classmates stuff above), but not really proof of anything, as far as couples are concerned. They (these moments the shippers drool over, like the Asuka stuff or the Rei stuff) are always about availability of choice, an availability that ends up being rejected together with the choice because relationships suck balls and get you hurt ("hedgehog dilemma" again): "I hate everyone and want everyone to disappear" cue orange juice with no glass or straw. Ta dah! The narrative dictates that any potential pairings are there just so there can be something to shoot down.
So much for your couples. :P
edit: oh yeah, and what I was going to suggest. If the article on Shinji and Asuka relationship is to stay it should be properly edited and similar articles focused on his relationship with Gendo, Rei, Misato and Kaworu should also be written.
another edit: I completely forgot about Yui. An article about Shinji's relationship with Yui wouldn't hurt either. She is his mother (and the fiend he pilots) after all.