A rather general question, isn't it? But alas, the most common one.
To summarize, when Ayato "tuned the world," what he did was recreate the world to which ever way he desired. But more precisely, he recreated the world as it was meant to be --he united both Mu and Human worlds. When Bahbem was meddling with the RahXephon system, he ended up splitting the world in two, one side were the humans and the other world were the Mulians. Without the history of the world split in two, many of the problems present in the series, including the time shift between Tokyo and the rest of the world as well as the Mu war, would have never happened.
In the end of this series, we see that Ayato and Haruka are actually together and married, but the very peculiar thing is that Quon is their baby. What this symbolizes is that both types of humans were merged together as one. The world is no longer discorded; no longer out of place in both time and race. In other words, there's no people with blue blood, or people with red blood (maybe purple? Or perhaps all red with the Mu-Phase like Ayato? Who knows.), there's no time alteration like Tokyo Jupiter with people being a lot younger than they should be.
Also at the end, we get a glimpse of characters that were killed in the series are now alive and well. It cannot be answered with 100% certainty just who is alive and who is not in this new world, though anything can be speculated. My educated guess is that everybody who was close with Ayato got a second chance. At least the people that gave Ayato a final message through the TV in the last episode for sure, which include Asahina, Mamoru, Souichi, Elvy, Megumi, and of course Haruka. Perhaps it can be considered a strong hint that Itsuki is also alive when he attempted to reach Quon by phone, giving her his last message.
That's Ayato!
The reason why Ayato looks so much like Itsuki, is because Ayato and Itsuki are actually twin brothers (Ayato still keeping the older brother status). How are they twins you say? Itsuki explained everything to Haruka in one of the later episodes in the series; he said that both him and Ayato are actually clones from the Bahbem Foundation... though they are indeed superior clones to the mass produced ones (the D's). They were made from genes taken from Quon (while she was in her stasis), and Watari Shirow (who most likely volunteered). The reason Bahbem created them is because previous clones using his own genes weren't functioning. So he had to fuse a product of somebody purely Mu and a human to make his experiement work or balance out. When Rikudo and Watari found Quon and Maya at the Shrine of Time, it was like hitting the Jackpot, since Bahbem was searching for a Mulian all those thousands of years he was establishing the Bahbem Foundation. When the two boys were born, Ayato was given to Maya to take care of, and Bahbem took in Itsuki to raise in his mansion along with Makoto and Helena.
Some speculate that Itsuki was merged with Ayato in the end since they're both twins. But I have a hard time trusting this. Not only would it shatter Ayato's character (as well as Itsuki's), but there's really no evidence of it. We are given hints that Itsuki wanted to be with Quon always, but that could very well be a brother: for real this time, instead of a father like Ayato is. Not only that, but Ayato also kept his name Ayato. If he were truly merged with Itsuki, I'd guess he'd have a mixed name or something, or we would be given stronger hints of this merger. I strongly believe that's just Ayato in the end... Haruka didn't even love Itsuki.
She was just playing around, probably wanted to hear the story again. Now, if the question were "Does Haruka remember all that happened in the anime?" then I'd say that's left for interpretation.
While I'm betting that Ayato does in fact remember (the golden RahXephon and Mishima paintings can be considered proof), I honestly don't know about Haruka or the rest of the cast. It's up for debate. She did took an important role in the tuning by helping Ayato stabalize his mind and making him realize his true goal, among other things, so there's a good chance she could have retained her memory.
As you know, it takes two RahXephons and two Ollins to tune the world, in the anime we see Ayato's RahXephon and Quon's RahXephon tuning the world. You could say Ayato was fighting on the human side, and Quon on the Mulian side, but truth be told, Ayato was most likely fighting for himself and for his loved ones; Quon was just leading the way to Ayato's victory because she also wished for a world where Ayato would be happy. The way the fight works, is that one of the RahXephons must consume the other to use that power to unite the world; to "tune" the world the two RahXephons must merge.
Ernst von Bahbem had been waiting THOUSANDS of years for THAT specific moment! THOUSANDS OF YEARS!! He has manipulated humanity, he has gone to the extremes of science, he has changed God knows how many bodies to stay alive, he has cloned humans to achieve his dream. Futagami, still keeping his sarcasm, saw to it that Bahbem didn't see his tuning after all he has done. Bahbem has simply forgone too much sin to be forgiven or compensated. Futagami saw to it that Bahbem is punished appropiately.
The Earth Federation has been keeping contact with Bahbem for a while. They were aware of Bahbem's power and influence and were very suspicious of him colaborating with the Mu. Terra has been acquiring too much power and the technology was much too similar with what the Mulians had. They sent Leutenant Colonel Jumonji Takeshi as an undercover spy (reporter Futagami) to investigate Bahbem's works, and if he's found out he's been working with Mulians to kill him. What Lt.Col. Takeshi found out was much worse. He finds that not only has Bahbem been working jointly with humans and Mulians, he's a Mulian himself! Futagami/Takeshi eventually puts it all together and even finds further proof with the speech Bahbem was giving him at the end; he finds out all the crimes againts humanity Bahbem has been doing. Lt.Col. Takeshi then decides to carry out his orders. He took Bahbem's literally life long dream with one shot... ouch! He figured he wouldn't give him the pleasure to see his dream come true after all the wrong things he has done to get it.
At this point in the story Sayoko had lost her mind. From the moment Itsuki found out that Sayoko had been cooperating with Makoto about the secrets of Quon, Sayoko has been since then roaming around discontent. We see scenes of her in the beach completely filled with grief, and a certain scene where she flipped on Ayato. From that moment on her sadness, disappointment, and perhaps even anger has been accumulating. It all exploded when she saw Itsuki one last time while he was holding a gun to Bahbem/Helena.
I can only conclude that she figured if she can't have Itsuki for herself, then nobody else will. So Sayoko, at this point it is apparent she's out of her mind, stabbed Itsuki. Once Sayoko realized what she came to a startling realization of what she had done. After this we get to hear Itsuki's last few words, and I have to say they were suprising since they were words of kindness and regret. We get hints that if Itsuki were to repeat it all again, perhaps he would have given Sayoko a chance. It is quite clear that Sayoko was deeply in love with Itsuki.
The real suprise was that Sayoko was yet another one of Bahbem's clones. More so, we are told quite clearly that she was "programmed" to love Itsuki, though her iradical behavior in killing him wasn't programmed at all. Why was she created for Itsuki? We can only guess. My take is that she was designed to give comfort for Itsuki, and perhaps even keep him busy. But when all is told, this is one of the more tragic endings in the series. It is not known if either Itsuki or Sayoko are alive and together after the tuning, but we can certainly speculate.
This is one of the most interesting ones. Even I was fooled with this one! You see, there are no Helena clones, there's only ONE Helena in the anime. That little girl that looks like Helena, the one that seems to always hang around with Bahbem, is nothing but an illusion occurring inside Helena's mind. It's Helena's sub-conscience telling her that she's also nothing but another of Bahbem's puppets. How do we see this? If you remember the scenes where Helena is talking with Bahbem, and little Helena is in his laps, there are a few instances where Bahbem is cherishing mid-air... o_O What does that mean? It means there's nothing there!!
Let me explain, as you know, Helena is a very proud person, in fact, she's extremely proud, if you remember, she was a spoiled brat as a child. So having her diminishing to the level of a disposable clone was a great insult to her, and more so, it screws with her head! (Though I'm betting it's Bahbem screwing with her head) So the little Helena inside the real Helena's mind is suppose to represent her as Bahbem's dear niece, but it also represents that no matter how powerful she can get with her politics and positions, she's still under the grasp and care of Bahbem.
And as you saw in the anime, Bahbem's purpose for Helena was to use her body as his own (trade bodies, so that Bahbem keeps living on). Why? Because women get less trouble going around making deals (men fall for hot women... that's a true statement) and what better way for Bahbem to deceive men than using a bit of sensuality. But of course, we saw how Bahbem ended up.
This question is referring to the scene where Haruka takes a jet and flies directly in the middle of the RahXephon fight. Why did she do this? There could be many reasons. A few moments before this scene we see that Haruka was about to give up, that apparently she had lost Ayato "forever" this time. But Megumi's words of encouragement drove Haruka to try anything to get Ayato back. So the most logical thing to do was get a jet and fly directly to Ayato and establish direct contact. Death was certain, but Haruka had to give it a shot. It can be speculated that either Maya brought Haruka to the after-world or Ayato. We're not really explained how Haruka arrived there. But one thing is for sure, Haruka got to meet Maya and get her blessing and relive an important memory with Ayato. If it wasn't for Haruka, Ayato would most likely have never realized his true goal for the tuning.
As sad as it is, when Ayato merged with the RahXephon, he simply lost control. When he merged with the RahXehon, he became a sort of god. He's no longer human, and cannot go back if he wanted to. As a side-effect, Ayato lost control over his power and ended up killing everybody without even thinking. In other words, he wasn't conscious when he did this.
The purpose of the Bahbem foundation was to create the artificial being known as the RahXephon, and fund a new organization, Terra, to take charge of any problems dealing with the Mu. There's a rich history behind the organization, you can read it in detail in the Timeline, or read a shorter version in FAQ #2.