I haven't put any photo of the case after I cut it, so voilà :

And here with all the components connected with each other.

For now I have a few things to fix, to go with a completely working Gameboy, like cut a bit more the case (1mm) to fit the battery, some system improvement to center the screen. Then I would like to desolder the LEDs from the powerboost to put them in the hole near the screen, and get a feed back about the battery.