Here are the answers to your lore questions ( spoilers below for Happiness 1 Definitive Edition)
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1- Mirall arrived to the timeline to find Butterfly and stop him. She was there for a few months by the time she went missing. The reason she enrolled in Sandy High school was to keep an eye on her parents and make sure they are safe and also to make sure they end up together without her directly intervening.
2- A time branch is where millions/billions of timelines converge together regardless of their point of origin.
3- This game has 2 T.S.R . the first version that Mirall used was created by the future version of the MC. The second version was created by Butterfly.
4- The thing about Mirall meeting the MC before and him losing his memory was covered in a web story I wrote back in 2018 called '' Happiness 0'' here's the link if you want to read it:
https://pdfnbooks.blogspot.com/2021/11/happiness-0-prequel.html
5- You can read the other web story called '' Happiness 1.5'' by simply downloading from this itch page
Thank you for supporting the series and I hope you enjoy the remaining games in the series.