Here are some answers to your lore questions (Spoilers for Happiness 1: Remake)
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1-Time branches are basically where a huge amount of timelines converge. In your example if you wake up in the morning and have a choice to either go to school or go back to sleep this creates 2 timelines ( Timeline A and Timeline B) however you have to keep in mind that you are not the only human presented with this choices, millions if not billions of people in your timeline get this choice too which creates billions of variations and timelines.
These timelines constantly converge and diverge the moment another choice is presented to you and every other human in your timeline.
2- In Happiness 1: Remake Butterfly is in his late 40's.
3- Mirall does know Butterfly, she been trying to stop him from destroying time branches for a while before the events of Happiness 1: Remakes.