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.

Спит is the 3rd person singular present tense of спать (to sleep).

So:

• я сплю = I sleep / I am sleeping
• ты спишь = you sleep / are sleeping
• он / она / ребёнок спит = he / she / the child sleeps / is sleeping

In Russian, the present tense often covers both English simple present and present continuous, depending on context. Here it clearly means is sleeping.

Лучше не шуметь is an impersonal way to say it’s better not to make noise.

Structure:

• лучше = better
• не шуметь = not to make noise

This is a very common Russian pattern:

• лучше подождать = it’s better to wait
• лучше не спорить = it’s better not to argue

If you said не шумите, that would be a direct command to more than one person (or a polite singular):

• Не шумите! = Don’t make noise!

So the actual sentence sounds softer and less direct. It gives advice or a general recommendation rather than an explicit command.

That is normal in Russian.

In лучше не шуметь, there is no explicit subject, but the meaning is understood from context: people in general, or we/you/anyone nearby should not make noise.

English often also does this:

• Better not make noise.
• It’s better not to be loud.

So Russian does not need to say нам or вам unless the speaker wants to make it explicit.

Шуметь means to make noise, to be noisy, or to make a racket.

It is an imperfective verb, used for the general action of making noise.

Examples:

• Не шуми. = Don’t make noise.
• Дети шумят. = The children are being noisy.
• В коридоре шумят. = People are making noise in the corridor.

In this sentence, не шуметь means not to make noise.

Because this sentence expresses location, not motion.

• в коридоре = in the corridor / in the hallway
• в коридор = into the corridor

Russian uses:

• в + prepositional case for location
• в + accusative case for movement toward somewhere

Here no one is moving into the corridor; it is simply the place where the noise should not happen. So в коридоре is correct.

Коридоре is the prepositional singular form of коридор.

Base form:

• коридор = corridor / hallway

After в when it means in (location), Russian usually uses the prepositional case:

• в коридоре = in the corridor
• в доме = in the house
• в комнате = in the room

So the ending -е shows the prepositional singular.

Yes, but only a mild one.

The sentence is: Раз уж ребёнок спит, лучше не шуметь в коридоре.

A very natural English-like breakdown is: Since the child is sleeping, it’s better not to make noise in the corridor.

The clause Раз уж ребёнок спит comes first because it gives the reason/background. Then the main point follows: лучше не шуметь в коридоре.

Russian word order is flexible, but this order is very natural because it presents:

1. the circumstance
2. the recommendation

Yes, but the tone would change a little.

• раз ребёнок спит... = since the child is sleeping...
• поскольку ребёнок спит... = since / because the child is sleeping...
  This sounds more formal or bookish.
• раз уж ребёнок спит... = since the child is sleeping / given that the child is sleeping...
  This sounds more conversational and slightly more like well, given that this is the situation...

So раз уж is a natural spoken choice here.

It is mostly neutral conversational Russian.

Nothing in it is rude or slangy.
The phrase раз уж makes it sound a bit more spoken and natural in everyday conversation.

So it would fit well in ordinary speech:

• at home
• in a school or daycare setting
• in a building or apartment hallway

It is not especially formal, but it is perfectly normal and polite.