Hidamari Sketch/WMG

Miyako is not originally from Fukuoka Prefecture.
Miyako's family probably came from some other part of Japan (the Kansai region seems possible, as one time, Yuno and Miyako were recalling that Sae called their teacher "Yosshi", and wondering what she'd be called this year; Miyako came up with something that made Yuno ask "What's with the Kansai-ben?"). Rather than travel by car, van, bus, or airplane, they took a ship for whatever reason. Unfortunately, during a storm, it hit something that made it sink. "There weren't enough lifeboats," and Miyako was forced to swim for the first time, or die trying. She came ashore on Kyushuu, perhaps knowing roughly where her new home was, and knowing that Fukuoka was in the north, she had to rely on the stars to guide her there. (And what's the direction that the stars will most readily point you in, in the Northern Hemisphere? Sure, you can derive the other directions, but...) She was eventually reunited with her family.

Natsume actually wanted to be more than just "best friends" with Sae, as the manga seems to say.
If she merely wanted to be friends, hey, that's no problem; she could've just joined the friends she already had. But Natsume wanted something more, and Hiro's apparent relationship with Sae gets in the way of that. This is why she took it so hard, and became something of a Jerkass to her. (On a related note, she seems to be afraid of Hiro alone, for whatever reason. It's not like Hiro has ever been intentionally mean to her, that we can see, and Hiro doesn't exactly evoke feelings of fear; perhaps Natsume's brain can't handle the contrary feelings of extremely nice person vs. winning rival for Sae's affections, so rather than snark at the nice person or be friendly to the rival, neither of which she can do, she gets out of there ASAP.)

The Nutbladder is real.
We just need to look harder.

Miyako's family name is "Maya".
In the Sports Fest episodes (one in which a typhoon keeps it from happening, and the other, in which it happens), when Miyako is called up to recite the athlete's oath, whoever calls her says "Maya" rather than the pronunciation "mee-ya" that we know from all the other times she's named. I had thought it was a mispronunciation, until I realized they probably wouldn't use her given name for an occasion like that, even if they were to call her Miya-chan when speaking to her one-on-one. (Doesn't explain the lack of honorific on this occasion, though.)

Family names don't exist.
Every character is called by only one name because they only have one name.