Why would anyone who knows Gray and the kids to not treat "sleep with" as literally just sleep with?
Sleeping in the same bed is still fairly intimate. Not necessarily
that kind of intimacy, but still intimate. Of course, in this case it's
familial intimacy, a father letting his younger children sleep with him and the eldest daughter joining the others to help out.
I'm not sure why Emilia is so self conscious about someone who is, for
all intents and purposes, is his daughter. The only way I can think of it making sense is if it has something to do with her
own self-perception, or perhaps some form of unusual anxiety? I hope things are cleared up soon (or that she just takes a look at herself and reprimands herself for being nervous over a
child) so that Emilia can become and larger and larger part of their lives. It seems like it's already going in that direction, so I'm hopeful.