Obscurity ahoy, but Scott Westerfeld's YA series is full of Fridge Logic, much of it based on the Great Big Book of Everything - the ten thousand year old 'lore' of the series. While it makes for a pretty cool and entertaining series... well.
Among other things:
An hour is not a universal measurement. Our 'hour' is based on the Babylonian numbering system. The Romans, for instance, had twelve hours from sunrise to sunset - not hours of regular length - and four watches at night. What are the odds that a period of exactly an hour, which almost no one experiences, has existed for ten thousand years?
Likewise, longitude and latitude aren't universal - they're based on the Babylonian too. So why does Bixby, the Landmark of Lore, just happen to fall exactly on ominous degree marks?
For that matter, thirteen isn't a universally unlucky number. Four is equally unlucky in East Asian cultures (if not more so), but receives no mention. For that matter, the series seems to assume a base ten numbering system, when the nearest culture with writing - the Aztecs - used a base twenty system.
I would think this, as well as the two previous points could be explained by flipping the causation. If we assume that the blue time was at one point accessible from other places (either it shrank over time, since it is mentioned to be spreading outward at the end of the series, or there are simply other places on the planet where it happens), then it wouldn't be too far of a stretch to claim that the use of hours, the development of latitude and longitude, and the "unlucky thirteen" beliefs came, in part, from the blue time. Babylonians midnighters develop a system of time that fits with the blue time, beliefs in the badluck of thirteen came from it's relation to darklings... it's not a very likely explanation, but it is possible.
It's possible the darklings decided to take exactly one hour because it's 12x12x25 seconds (leaving us with 12x12x12x50 seconds, and creating a 24:1 ratio) because those were the types of numbers that they could mentally deal with. Though nothing says the base unit for time needs to be seconds.
Causing time to stop and everything to freeze until touched: very common in time manipulation, but if everything is frozen, how does sound carry? (We'll chalk 'how do they stay warm' up to magic.)
Midnighters and darklings are subject to special rules (magic, basically) which let them function without apparent problems in the blue time, which explains why they stay warm. The fact that midnighters can touch small things (coins, leaves, rain) that were caught in mid-fall and can make them move again means that these rules extend to the parts of the world they influence. Flame bringers presumably have a larger sphere of influence, which is why they can make machines and such work.
The series says that new human inventions, like fire and numbers, began to take effect about ten thousand years ago. Humans have been manipulating fire for at least ten times that long, and numbers for about four times as long. It was around the period of the Fertile Crescent, which may have influenced the matter... except that the series takes place in Oklahoma, which didn't have farming then, and all of the lore is local.
The local polymath, who's supposed to be incredibly amazing at math... is only accelerated by one grade level. While she certainly seems to know the material, she seems to have a mental calculator, not incredible intelligence, because she only uses her power for counting, arithmetic, and unspecified, semi-instinctual graphing.
That's what I think on the polymath power. She knows how to do the math, just not what to do with it. The only practical applications of math that the polymath power comes with are Midnight-related, so she needs to go to math class to learn how to use it in the normal world.
Alternatively, it could be either a "blend in" thing - she'd stand out if she was advanced more than one grade level - or else the city's/school's policy that no student can be more than one grade level up in any class.
Kind of small, but something that comes up in the first book was how they told time during the secret hour by the position of the dark moon. Yet in book 2, Rex can use a watch during the secret hour, and then in book 3, it's Hand Waved that wind-up watches work for all midnighters.
In the first book they do mention that wind up watches work during the secret hour. Still bugs me though. Shouldn't the watches have stopped moving like everything else? Why don't all the other wind up clocks keep on ticking through the secret hour too?
Most things that a midnighter touches (excepting more complicated chemical reactions, like fire, and electrically based items, like digital watches and flashlights) stops being frozen in time, so wind-up watches should work.
The Blue Time exists between midnight and 12:01, and time is frozen for an hour...which shouldn't happen because 60 seconds pass between 12:00 and 12:01, so why would time completely freeze? If anything, time would keep going, but everyone else would move slowly compared to the midnighters, who are experiencing more time. If there was to be an extra hour, than it wouldn't be between one minute and another, it would just BE, and no normal clock (unaffected by the midnighters) would show any passage of time, whether it be a minute, a second, or a millisecond, once the Blue Time ended.
It's stated that the secret hour passes in a single instant, not in the entire minute between 12:00 and 12:01. Although, not every clock will say exactly midnight because not everyone's clocks are set perfectly to one another.