Yeah, Minami's dad was an abusive drunk who came after her with a knife.
Note that this would not be incompatible with the parental murder-suicide story Shizuka describes (though realistically she gave her own horrible backstory to establish shared ground.)
No, but it would be a bit of an odd narrative choice to leave "she buried the bodies of her parents" out of the earlier infodump about a traumatic backstory.
Shizuka found out, intending to lord her backstory over her, but realized she sympathized, so she is sharing her own backstory because she feels like Minami can relate.
That's a pretty debatable reading, as her self-described motive for prying rather appeared to be genuine curiosity about Minami...
She explicitly says it was "to keep the advantage" though. The next time she sees her, though, she immediately gives up that "advantage" by sharing her own background
Curiosity is what drew her to her in the first place, but she explicitly stated her reason for stealing the answer was to have an interpersonal advantage over Minami.
So holy shit. Shizuka's mom pulled a murder suicide and Shizuka had to bury the bodies. That is so dark.
Goes ways to explain why she's so chronically fucked up, certainly.
On another note, was I the only one to notice that Shizuka's version (recounted internally ie. without any need to gloss things for the benefit of the target audience) of their initial acquaintance indicates the one Minami gave Iori way back in ch 12 is basically BS? Not that it's hard to see why she'd prefer to tell a take that's both rather compressed and "lighter and softer"...
It does, however, have the unfortunate side effect of rendering anything Minami tells about her past inherently suspect at least as far as details are concerned. :/
The story Minami gave wasn't about how they met, it's how they interacted on a daily basis. We don't really have any reason to doubt that both stories are true.