Alternate Universe Fic: Evangelion R takes place before End of Evangelion and overlaps with it, timewise. Due to various changes in the timeline, End of Evangelion doesn't happen, though many of the events from it still occur, albeit altered to some degree.
Apocalypse of the Week: Notably averted, considering the source material. Tokyo-3 is only attacked a handful of times in the pretty lengthy series, which instead focuses even more on character drama than the original show did.
Break the Cutie: Played straight, averted, and subverted alike. While Lyn and Aoi both go through some pretty serious breaking (along with mainstays Asuka and Shinji) they may come out of it alright (depending on which ending you're following).
Bridge Bunnies: When Maya gets promoted, she's replaced by another Bridge Bunny.
Shinji: Why can't you just open up to me? Asuka: Because... because ich liebe dich, you jerk! Shinji: Wh-what? What does that mean?
Also occurs to both Aoi and Lyn, who for most of the series go in circles around one another.
Character Development: Most, but not all of the characters, receive some development. Notably, Lyn and Shinji become more confident, Asuka less bitchy, Rei more in touch with her emotions, and Seyoko more responsible (as well as Misato too, to a lesser extent).
Conspicuous CG: The trailers. They don't look bad considering they're fan-made (and from the early 2000s too) but they certainly don't look much like the original series or the few traditionally animated clips they throw in.
Contemplate Our Navels: Like the original, very much so. The main difference is that now Shinji shares this duty with Aoi and Rei, who do most of the navel-contemplating this time around.
Tenkei gets in on some of the navel-gazing action as well.
Continuity Nod: A few. For instance, recalling the "personal hell" sequences from the original series that so often involved trains, Shinji, just before departing on one for a school trip, remarks that he loathes trains.
Cyberpunk: Played up a little compared to the original. Notably, one story involves using the technology that allows Evangelion synchronization to try and cure blindness.
Dude, She's Like in a Coma: Stronglyaverted. The second scene in the first episode is a deliberate mirroring of the opening for End of Evangelion, where Shinji engaged in his... erm... appreciation of Asuka. This scene clearly sets up the different tone the fic aims to take with its character relationships.
Gambit Pileup: Dear gods, yes. First, there's Seele plotting against the world in general, as in the original. Then there's Gendo plotting against Seele, which naturally they know about, so they plot against him. And then there's Misato, who's plotting against both Seele and Gendo, in which she is eventually assisted by Seyoko, who was independently running her own continuation of Kagi's schemes, who is herself part of a plot by Seele and UN Synaps. Oh, and then there's Aoi, who's plotting, depending on which ending you're reading, against the world or against SEELE. And Ritsuko, not to be left behind, get in on it a little too, plotting against Gendo in particular.
Glomp: Aoi does this to Lyn occasionally, much to his embarrassment.
A God Am I: Aoi, as well as, to a lesser extent Tenkei
Gratuitous Foreign Language: From time to time, often in imitation of the original series with Asuka. Occasionally, Tenkei also speaks what seems to be Arabic and is implied to be Babylonian. Oddly, because the series was written in English instead of Japanese, Japanese serves this purpose as well while Lyn's dialogue is implied to be Gratuitous English but, of course, isn't.
The Heart: Maya in some ways and Aoi in others. Maya certainly serves as the most innocent of the cast, giving a moral center to the series. But it's Aoi who's constantly cheering people up (or at least Lyn).
Incest Is Relative: Averted, as Shinji's attraction to Rei is implied to have come to a very abrupt halt following his discovery of her true nature. Made worse when Rei recovers some of Yui's memories.
Kavorka Man: Averted as neither Shinji nor Gendo have the harem they had in the original series. In Shinji's case, Misato matures and stops flirting with him, Rei grows apart with him and bonds with Tenkei instead, and Asuka comes to grips with her feelings for him. In Gendo's case, Ritsuko plots against him while none of the other females seem to show any interest in him at all.
Kid Hero: Reconstructed, as much of Shinji's character arc is him coming out of the pit of despair he'd fallen into at the end of the original series. Still deconstructed with Lyn though.
Lethal Chef: Misato's continued inability to make food the rest of the household considers unpalatable becomes a minor plot point in one Bottle Show, in which Shinji is made the designated cook.
Lighter and Fluffier: To a certain extent. A lot of angst and disaster remains in the series, but Shinji grows closer to his teammates and comes out of his shell while newcomer Aoi is, while not without her issues, a pretty normal kid by Evangelion standards. Only she isn't.
Mega Corp.: UN Synaps, sort of. They're actually (as their name indicates), a branch of the United Nations, but they act in a way that is very reminiscent of a Mega Corp. and are treated as such within the context of the story.
Myth Arc: Very much so, and with much more details than the original series. Whether that's a good thing or a bad thing depends on your point of view, as it can also reduce the mystery that surrounded the original Evangelion.
Oracular Urchin: Like Rei, Aoi appears to live without a family. Like Rei she knows a lot more than she's letting on.
Our Angels Are Different: Surprisingly, yes. First of all, there's the fact that the angels which attack Tokyo-3 during the series are nothing of the sort. Then there's Aoi, who much more closely resembles the popular depiction of angels than most of her kind in Evangelion.
Red Oni, Blue Oni: With Rei off doing her own thing, Aoi fills the blue oni role against Asuka. If anything, they seem to loathe one another much more than Asuka and Rei ever did. This is probably in part because Rei wasn't exactly a very hateful person.
Suspiciously Similar Substitute: Though Shinji ends up sending much of the story growing out of his insecurities it seems the author decided he still needed an Extreme Doormat. For this purpose, we arrive at Lyn - the British Shinji.
What Measure Is a Non-Human?: Tenkei. He's treated as a thing by NERV, even more than Rei was (who Gendo seemed to have genuine affection for). However, he's clearly capable of emotion, forms a close bond with Rei, and is a clone of Kaji. On the other hand, he is a weapon of war and has quasi-deific rings to him.
Yandere: Aoi does take rejection well. She gets over it though. Ritsuko does this a little as well, though only at the beginning.
You Gotta Have Blue Hair: Oddly averted with Aoi, who not only has a Meaningful Name but who is also an angel, like Kaworu. She just has regular black hair. She does have blue eyes however which, while strange, is not unheard of in Japan.