♥ Where are the actual parents in all of this? Who's paying Hatsune's hospital bill?
There is nothing I can use in the source material regarding this. Therefore, I'm going to take creative license here, but I will amend anything that gets light shed on it. IF it ever does.
Hatsune's illness is never explicitly stated, though cancer is a popular theory. However, Yuzuru states that she has only been hospitalized for the last two years, not her entire life. She is clearly older than two, meaning that the illness would have to be dormant up until two years ago. This could mean a genetic disease that is active in her, but not in Yuzuru, which would contribute further to his guilt. He expresses wishing it was him who was sick, not her. Because Yuzuru lives alone in a house but is sent no support from parents, it's probably safe to assume that Yuzuru and Hatsune are orphaned. Their mother may have died due to complications after delivering Hatsune or from the same illness, or the illness could have come from their father's side. Either way, the immediate Otonashi family consists of two orphans, with no monetary support and nothing but a home and whatever survivor benefits they have: which all go to Hatsune's hospital stay.
♥ Anything for Hatsune.
Just what it says: he would do anything for his sister. This is proven. Perhaps she learned to braid by using his hair, or he assisted her by practicing often, which then lead to her characteristic small braids; maybe he sat through shows that she liked but he wasn't a fan of.
♥ Not just teen angst.
As a younger teenager, Otonashi is absolutely pessimistic about life. He questions the point of being alive and wishes he could trade places with his little sister, effectively saying he would rather suffer and/or die in her stead.
I think it's safe to say that the culmination of saying life is pointless, knowing his baby sister is dying, an apparent lack of parents, and choosing to avoid school ("I never go, I hate pretending to fit in, etc") in favor of working in hard labor might lead to more than just teenage angst, but legitimate depression. Potentially even depression to the point of considering suicide (which, let's be fair, isn't uncommon in teenagers). He has to stop himself from pursuing this line of thought during one of his inner monologues.
He had zero friends, often skipped classes entirely, and may very well have been bullied. By his own admission, he preferred to play video games alone in his room. That alone is not a sign of depression, but with everything else surrounding it, it's very probable that he was. Furthermore, without Hatsune's affection, he may have entirely given up.
♥ Pre-death school life.
This builds on the above, essentially. He was probably harassed. He was not terribly masculine or confident; instead, he sold himself short, saying he was "dumb" when school is for "smart people". He slouched, struggled to fit in and disliked it when he had to. This kind of behavior is practically a magnet for school-age children and teenagers, since it denotes a vulnerable peer.
Adding to that, he worked minimum-wage labor jobs in a grocery store and as a traffic conductor, where he was consistently getting yelled at. This is only shown twice, but anyone who has worked one of these jobs knows it can happen often. Customers can be unnecessarily cruel and Otonashi was young, inexperienced, and often making mistakes. It's no wonder why he would question the point of his life.
♥ Disaster response: 7 days to save the survivors?
During the train wreck scene, Otonashi and the other surviving passengers have no signal and no way to contact the outside world. In modern-day Japan where this occurred, a crisis like this would have been taken care of in a timely manner. When a mine or tunnel collapses, rescue teams are on it immediately.
Both of these scenarios are isolated events, thus resources can be immediately devoted to them. What happens in the event of an earthquake or a tsunami? Japan is located on a triple fault line and is therefore susceptible to said disasters. An earthquake that took out the area would have delayed a response team. This might have left the passengers underground for a while -- even an entire week.
Also, the fact that both ends of the tunnel collapsed at the same time is curious, but possible in the event of a disaster.
♥ Otonashi and going underground.
Yuzuru died underground after acting as a leader and moral support for the few remaining survivors. When the SSS begins their descent into the guild, Otonashi hesitates. "It's... in there?" he asks, lingering behind everyone else except Hinata, who then nudges him towards it. Notably, Hinata ends up heading down the ladder first even though he was trying to urge Otonashi to go ahead.
Hinata is naturally friendly and welcoming to almost everyone, but he is extremely watchful of Otonashi here. Was Otonashi giving off more of a lost vibe than usual? Even if he does not remember how he died at this point, he may be wary of going underground with good reason. He is shown to pick up old, non-specific memory when he is reminded of it so this instance could be similar. For example, he knows how to do physics, but he couldn't tell you why he knows or why it's so easy.
He might be hesitant to enter the underground without knowing exactly why.
It's also possible that the idea of a guild being under the gym just sounded weird to him. Why down there, huh? Still, I think it's something interesting to explore, just as AB! often re-references things about a pivotal character's past and their current behavior.
The Angel Beats! visual novel was recently announced with a release date of Spring 2014 with Otonashi as the protagonist. It was stated that his personality might be re-imagined depending on the choices the player makes. This is standard visual novel fare.
It was also stated that he has routes with any character, male or female (though it is not clear if they're all romantic in nature). According to the information that's out, the player can alter his personality and sexuality based on their choices. Therefore, he can be straight, gay, bi, or perhaps disinterested in romance (or just completely asexual). What does this mean?
Basically if it seems like he "fits" with someone and he's in the right place in his development for romance, he may go for it regardless of gender and regardless of the original anime's canon pairing. That's all.
♥ A sister complex.
Given the information on Otonashi's history with his terminally ill younger sister, it is safe to say that he was protective of her. His love for her drove him to continue living and, eventually, to work hard enough to take her out for a memorable night over the holidays. Her death results from his good intentions, and I'm certain he feels a degree of guilt over this. I don't believe her death was directly his fault, but I do think his act of taking her out hastened the her inevitable demise.
It is not an uncommon occurrence for a person to want something they have lost. He clearly loved her more than anything else, considering these two siblings only seemed to have each other. I definitely do not believe their relationship was incestuous, but I do think it's feasible that Otonashi would have a stronger protectiveness and brotherly attachment reserved for young, small girls. He certainly respects their privacy, as he yells at the other SSS members for breaking into a girl's dorm room.
♥ Guilt surrounding Hatsune's death.
Otonashi does not come right out and say that he feels like he killed his sister, but he does admit to harboring a lot of guilt after she died. Hatsune did pass away as early as she did as a result of his actions. She might have lived a bit a longer if he had left her in the hospital like her doctors told him to.
I know it is not directly his fault and he knows this on some level too, but as humans we do tend to make these 'what if I had?' or 'what if I hadn't?' connections. He will not dwell on it all the time, but talking about little sisters will probably be a tender subject. Talk of sibling loyalty might trigger him too.
♥ Otonashi and Yui
Otonashi quickly resolves to find out Yui's past and help her find peace. Yui is a lot like Hatsune; both of them were young girls who were unfairly stricken with an ailment and left unable to ever live normally. Despite this, both Yui and Hatsune retained their outgoing demeanors. Otonashi realizes this and is determined to help Yui fulfill her dreams, no matter how ridiculous and impossible they seem. Though he gets frustrated internally, he doesn't seriously lash out at her and continues straining himself to help her.
He feels that he did very little for Hatsune, but he is absolutely determined to see Yui through. The idea that Yui's impairment = Hatsune's impairment can also apply to Kanade and account for his abrupt dedication to the happiness of both of these characters.
♥ Otonashi's last memory.
When he wakes up in the afterlife, Otonashi cannot recall anything specific about himself. He is shown later on to find a physics test "easy," implying that while he has no specific memories about himself, he can recall things he has learned or non-specific things he has done in the past. His mannerisms and skills prior to his death carried over into the afterlife. Since there was no need for a doctor's skill in a place where death is a temporary affliction, that subconscious knowledge never came to light, unlike the physics test.
As Otonashi was dying in the tunnel, Igarashi called him by name, trying to get his attention. This is the last thing Otonashi ever hears; when he tries to remember his own name in the afterlife, he can't call it to memory immediately. He might recall someone else saying it and it just feels familiar. Because nobody used his given name—Yuzuru—he only remembers the name Otonashi.