Does anyone know where does Emile go after she leaves the house?
Story-wise, she has her own place and works at The Institute. Gameplay, I don't think Sam Tail has made it that far yet. Sam Tail has said some things in the past about plans for Emile. These posts are old, but they'll give you an idea of what's happening.
Sam Tail:
"The idea is that Emile will stay with you for the first few days and show you around town. There will be a bunch of introduction events for new players that will show you lots of the more important areas and give some exposition about the area. There will also be more character building for Emile since she will be an important character moving forward. Then there will be an event where she leaves to go back to her own home."
^^^^^(This bit is done already)
"Emile has quite a large role to play in future events and one potential path is a lesbian route. My plans with her are roughly outlined here. There will also be other lesbian encounters where I feel it fits, but the majority of encounters will involve males."
"Emile moving in with you permanently is also a planned storyline. It will be triggered by the main quest after you find out that Emile works for the Institute."
"Unfortunately it seems a lot of the confusion stems from content that is planned but not in yet. The characters talk as if the content is already in, but it isn't so it's leaving you thinking you have done something wrong."