Questions & Answers about Раз уж ребёнок спит, лучше не шуметь в коридоре.
Раз уж is a conversational phrase meaning something like since, given that, or seeing that.
In this sentence, Раз уж ребёнок спит... means: Since the child is asleep... or Given that the child is sleeping...
It often adds the nuance of accepting a situation and acting accordingly. So it is a little more like: Well, since the child is asleep, ...
Because ребёнок is the subject of спит.
In Russian, the subject of a verb is normally in the nominative case.
Here:
- ребёнок = the child / a child (nominative singular)
- спит = is sleeping
So ребёнок спит literally means the child sleeps / is sleeping.
If you used ребёнка, that would be a different case, usually accusative or genitive, and it would not fit as the subject here.