Pg. 17
Yep, I think you’ve got it exactly right.
Remember, a ー in Japanese is different than an English -. It doesn’t indicate anything being cut off or interrupted, it indicates the sound before it being elongated. It could also be written as でさぁ, for example.
でさー is essentially very similar to だよね… It’s kind of a colloquial, “Right?” indicating agreement with whatever was said before. You’ll usually see it at the end of sentences, but it can be used at the beginning and middle of sentences. In the beginning of a sentence, it could either be a sound of agreement with what was said previously, or it could be a “continuation” of the previous sentence, “Oh! That’s right, and” sorta vibes to it. In the middle of a sentence, it’ll usually be a filler sound.
I know that one character in a mobile game I play uses it constantly as she speaks (the character is supposed to be from Okinawa, iirc), pretty much just as filler speech. “They were like this, right? And so I was doing this, right? And it was super cool, right?” would be the closest English example I could come up with. She uses it often enough that I read the でさー in her exact tone of voice because my brain is trained to hear it that way. 
That’s how I’ve seen it used. If there are other uses, or if there’s anything more in-depth to it than that, I don’t know it.
Pg. 19
Yep!
If you mean in the way that it’s a boilerplate/common response to bizarre things in anime/manga, I mean, yes, kinda? But it’s also just a casual と quotation particle + 何, so it’s also grammar applied perfectly normally:
“What (do you mean by), “Turned inside out?!” That’s scary, Kotoha!”
As for 出動要請, while it doesn’t show up on Jisho, it does show up in Weblio when searched, as one word: “Dispatch request.” I can also say it shows up as an option in my IME without me having to do anything special, so I’m fairly comfortable saying it’s supposed to be one compound word (It’s also a title of a song from the Fire Force soundtrack, for what it’s worth.
). Either way, the meaning of the sentence isn’t changed all that much whether it’s two words, or one compound.
“This is a dispatch request for us Colours, the protectors of this town’s peace!”