Well, I've got some basic unanswered questions, like her first name, date of birth, blood type, those kinds of things.
I've speculated that she may have been a loner during her first year at school.
She may have had friends in her swim club, but she's never seen talking to or about them, or hanging out with them. Since it's been said she's the best female swimmer in the school, maybe the rest of the club was intimidated by her? Maybe she didn't mind not having friends there, perhaps she'd rather focus on improving her swimming, as it's been seen that she takes it very seriously.
In that regard, it wouldn't be hard to see why her and Miss Kurosawa got on so well, not just since Kagura's the star pupil, but Miss Kurosawa might have felt that she needed a friend, or at the very least someone to talk to, which Kagura was happy to oblige.
After the initial shock and sadness of being beaten by Sakaki at the sports festival, she may have seen it as an opportunity to not only improve herself, but make a fresh start in second year and make some friends, under the tough girl act of making Sakaki her rival. The rivalry quickly became a friendship, due to Kagura wanting to be "Frenemies" anyway, and due to Sakaki's noncompetitive nature.
Once she became friends with the rest of the main characters, she's not once brought up her class from last year, or any previous classmates, which follows from the "fresh start" idea I mentioned.
This speculation actually made me relate to Kagura a lot more, since I was a loner for the best part of a year and a half back at school, before I approached a group of people who I felt I could relate to, and we've been friends for around 10 years now.
It's weird, I've had these ideas in my head for a while now, and I thought it'd make less sense once I'd written it down, but it seems more like the opposite.