Muichiro sees Mizue as an older sister figure that reminds him of someone he used to know.
Mitsuri, Mizue and Shinobu were supposed to do each other's hair the next day.
Rengoku asked permission if Mizue could come with him at their nightly hashira meeting. Muichiro, Mitsuri and Shinobu helped him with that and they were successful.
Muichiro hopes to go on a mission with Mizue one day.
Mizue sees Naho, Senjuro and Muichiro as her little siblings.
Iguro feels like Mizue and him have been friends for a while, but he knows they haven't so he's still a bit protective of the hashiras. (Especially for Mitsuri)
Akaza feels like a bad caretaker after not bringing Mizue home.
The reason why Aizetsu acted weirdly is because I headcannon, Urogi, Karaku, Sekido and Aizetsu as separate people :D
Naho sees Mizue not as a sister figure, but as a mother figure instead.