It's literally her own old house - it makes absolutely sense to find smth you lost there. She wasn't there since her parents died so she had no chance to search/find it there.
The friends already knew.
Also there was no need to "convince" her - it was an empty threat - her duty as a "mother", she said she wouldn't forgive/accept her/them if they continue.
I'm pretty sure she was trying to convince herself too with what she said. That’s also why she said it as a question because she wanted Mayumi to confirm/deny "her" thoughts cuz she isn't sure herself.
Also she doesn't know that this is a precious memory for the other party.
I think on page 16 is a mistake?
These are Kikyou's thoughts about Shinobu so it should be:
"She didn't understand, didn't know anything." "She was just a girl being manipulated by her surroundings."
Following with page 17 as you wrote: "She's really grown up now."