Why doesn't anybody bother to tell Yuuichi that his childhood friend who fell from the tree is still alive, albeit in a coma? That would have solved his trauma then and there—although it would admittedly have made for quite a bit less drama.
They were afraid of awakening past trauma. It's possible they wanted to talk about it, but kept putting it off.
On that note, what was exactly the Ayu from the taiyaki incident to her disappearance, and why is she surprised that the tree from which she fell was cut down, supposing she went there everyday? (EDIT: I don't buy the All Just a Dream WMG).
The Other Wiki says she's an astral projection. I assume she came to life in the same manner as Fuko, and that either she disappeared when she "went to school", or that she created fake memories for herself, truly believing she was going to her school, so she wouldn't have to face the hard truth that she was in a coma.
Alternatively, they may not have known much about Ayu (obviously not a solution for the 2006 anime, where Akiko explicitly knows about what happened, but in the original game Yuuichi learns that Ayu's still alive after hearing that she woke up from her coma on the news, IIRC)
Also, it's hard to claim Ayu is in the hospital when she's standing right there. Some of them may have suspected something, but decided they were wrong or that Yuuichi wouldn't believe them.
Why is the 2006 opening considered a spoiler? And by spoiler I'm talking about Ayu's pink wings. Are we referring to the angel doll Yuuichi bought for her? Because the wings on her backpack spoiler that more than anything else. Ayu isn't dead, just in a coma, so it's not like she's an "angel" communicating with Yuuichi beyond the grave. Yes, it does hint that she is quite important, but the series proper does this itself, and if you couldn't figure that out, than you're just not paying attention.
Why is it that Yuuichi gets over all the girls so quickly when they seemed so important to him? He kisses Shiori, marries Makoto, but only seems to angst about Ayu past the first episode of the next in the story.
Well, the obvious answer is that each story was meant to be taken as a separate work, and therefore any attempt to combine the two will result in some weird clashes between stories. I'll try to lay out a few thoughts on the matter, though.
First off, Makoto's story. I'm going to more or less agree that he does get over her death fairly quick (especially since the two of them got married-ish). On the other hand, the more cynical part of me that pops up whenever I find a romance that I enjoy points out that we don't know exactly how Yuuichi felt about her. I got at least a bit of the sense that Yuuichi married her because that was what she wanted, and wasn't necessarily interested in her romantically (this possibility will show up several times in my points, as it helps alleviate the problems).
I think this is probably close to what's intended. He had feelings for Makoto, but the "marriage" was him trying to make her happy for her final moments. After she died, and he made his wish that she would find a peaceful eternal spring, what else was there? There were some other mentions through the story (2006 anime, that is) that he missed her, IIRC.
Mai's story: Even in the original game, Yuuichi and Mai seemed more like friends than they did a couple. Presumably Yuuichi never considered her to be anything more than a friend. Yuuichi's still free to pursue other romantic interests because he never went after Mai in the first place, and he doesn't spend time worrying about her because it's already clear by the start of Shiori's arc that Mai and Sayuri are going to recover.
Shiori's story: If I recall correctly (and I may very well be wrong), Shiori kisses Yuuichi, and it's only a kiss on the cheek at that. The relationship in this story always seemed a little weird to me: Yuuichi's relationship with Shiori is possibly the closest he ever comes to having a "normal" romantic relationship out of all of the paths (with the possible exception of Nayuki's), but at the same time there are a few scenes which hint/state that Shiori considers him to be more like a brother. Slightly more problematic is that even then, he should probably care a bit more for Shiori after her path (does he ever visit her?)
I thought that a) yes, it was more like a close, caring, sibling-esque relationship than necessarily a romantic one, but also b) that Shiori was in very bad shape after her 16th birthday, and not really able to receive visitors until Ayu's miracle healed her. Obviously, the anime played it off as her having died to up the drama. But when they meet again in the last episode, I think their feelings for each other come out pretty well ("Yeah, it's okay to cry, because you've been holding it in all this time").
Nayuki's story: not a problem, Yuuichi is completely oblivious to her feelings.
He did suffer a near breakdown later in the series, so it might be that he was holding it all in, swallowing his grief.
Where's Ayu's parents? She seems to mention them, but no explanation is really given for what happened to them. Both for as a child, and for when she was in the hospital, did they just leave her for dead or something?
Ayu's father moved away from their old home, but he's probably still in the same town. It's just that, her condition hasn't really improved any after seven years, so it's only natural that his visits would grow infrequent over the times.
A much more definite answer exists with regards to her mother: she died shortly before Ayu and Yuuichi first met each other
In the VN, I assume based on the scrolling text that it is thanks to Ayu's last wish that Mai, Shiori and possibly even Akiko survive. Then why doesn't she use her wish to allow Makoto to stay with Yuuichi?
Was that how Mai survived? I thought she revived herself. As for Makoto, I assumed either she didn't really know about Makoto (unlikely, since she left before Akiko's accident, but still learned about it), or if she simply couldn't do anything about it. Akiko, Shiori, and Ayu herself were all alive, just grievously injured, terminally ill, and comatose, respectively, whereas Makoto was, in a way, already dead.