Why did Summer B wake up just as the ship was about to sink?
The 'ship' was actually their cyrogenic unit that had launched to the surface (from deep within the ocean) after its computer had deemed the circumstances right to awaken its inhabitants. This is why they woke up just as the ship reached the surface and appeared to be 'sinking' due to the storm.
And shouldn't the Government have to inform the participants in the Seven Seeds project that they are going to be cryogenically frozen?
Aside from Team Summer A and the guides, none of the Teams knew about the plan because they would never have agreed to leave their home, friends, family and loved ones and get cryogenically frozen, then wake up who-knows-how-many-years-later after the world has been destroyed and be forced to survive without any proper training beforehand.
It's mentioned in the various Apocalyptic Logs that in the immediate years after impact, contact between the shelters and manned Fuji facilities were eventually lost. This could simply be the result of faulty equipment or the communication lines decaying. But it's never stated outright that the other Japanese shelters were destroyed, leaving the possibility that there might still be a few out there that are actually surviving.
According to Mark's journal, the Ryugyu shelter was designed to have enough food supplies to last three years. Take a guess how long said shelter lasted until everyone in it died.
Subverted, since things seem to have fallen to bits within months. Mark's diary mentions that he came to the shelter during the summer. The food storage was originally stocked with enough to last three years, but a complication resulted in it being lessened to only last about three months. The final events of the shelter take place on Christmas, so it went from good to bad within six months, at most.
The ceiling of the Ryugu Shelter being used as a giant projection of weather, including various seasons and weather types. It may seem like a really small thing to waste time on creating, but thinking about it, it does lead to a slight sense of normality for the inhabitants of the shelter, making it easier for them to be stuck all the time.
Hana's survival skills. She was raised by one of Summer A's teachers, so she got the benefit of the same knowledge, but without the attendant trauma because Takeshi wasn't willing to inflict it on his own daughter after seeing the results last time.
Hana's upbringing in general makes a lot more sense in light of her father's history. Of course she's obsessed with finding Arashi, because he was her Only Friend as well as her boyfriend. And her social isolation makes sense too, because her parents tried to keep her from bonding with people who would just die anyway when the asteroid hit.
