Basically, exaggerated or 'posey' ways of expressing an emotion or an idea that no real actual person would ever show, outside people whose job it is to publically act like anime characters.

Oh, so you dislike their emoting and personality, not their 'animation' fine.

for example, with the 'crazy girl stare' they give Azami

Yes, evil robot character.

whatever the hell it is Hotaru is doing there.

Yeah, this one is a little hard without sound, but she's giggling to herself before turning to talk to the player.

Raiko for some reason is shown with a very unnatural looking cry-laugh face

She's not laughing? She's teary eyed because she just struck herself with lightning.

Uka is... jesus christ, her design is already 'furry bait', what's with the bum in the air scuffing about?

She's literally acting like a dog. Like you said, furry character.

but was that necessary?

No, but I guess her entire character is unnecessary. I don't see a necessity argument here. It was added to have more personality to her specifically.

What is this trying to tell us about the character that couldn't be conveyed with more nuanced animation?

Like what? They have like 4 seconds each. And I don't really agree with just insisting on 'more nuance' without much nuance to your criticism itself to be a meaningful complaint.

Several of the others show extremely little personality in their faces, they show off some glitzy moves, strike a cool pose, and that's basically it. Same question: what is this supposed to tell us about the characters?

Their general personality and ability set?

I also think you're being inconsistent here, by insisting that the other characters are too exaggerated in their personalities but the others being subtle and nuanced you seem to object to as well.

I think you're being weirdly focussed on the personalities of these characters. I mean I guess they could be better but anything could be made better. And so far these are such minor characters, that I'm not sure anything else is needed.