Learn Hardcore Norwegian: Sykepleieren ber de pårørende vente i et stille rom. - The nurse asks the relatives to wait in a quiet room.

sykepleier = a nurse (indefinite)
sykepleieren = the nurse (definite)
The ending -en is the common definite ending for many masculine nouns in Bokmål.

What verb is ber, and why not spør?

ber is the present tense of å be (to ask/request), and it often implies a request/instruction.

å be = ask/request someone to do something (can sound more formal/authority-based)
å spørre = ask a question / inquire
So Sykepleieren ber ... fits when the nurse is requesting/directing someone.

Is de here the same as the pronoun meaning they?

Here de is a determiner meaning the (definite plural), not the subject pronoun they.

de pårørende = the relatives / the next of kin
The pronoun de (they) would typically be used without a noun after it.

What does pårørende mean grammatically—why does it look like an adjective?

pårørende is originally an adjective/participle meaning roughly related/connected, but in modern Norwegian it’s very commonly used as a noun meaning relatives / next of kin (especially in medical contexts).
So de pårørende functions like a normal noun phrase: the next of kin.

Why is it ber de pårørende vente and not ber de pårørende å vente?

After å be, Norwegian often uses a bare infinitive (without å) when you have be + someone + infinitive:

å be noen vente = to ask someone to wait

You can also use an alternative structure with om:

å be noen om å vente (also correct; can feel slightly more “request-like”/polite or a bit more explicit)

What’s the role of vente here—tense, form, etc.?

vente is the infinitive (to wait). In this structure, it’s the action the relatives are being asked to do. It does not show tense itself; the tense is carried by ber (present tense).

Why is it i et stille rom and not på et stille rom?

Because you wait inside a room, Norwegian uses i (in).

i et rom = in a room
på is used more for on a surface or in certain set location types (e.g., på sykehuset = at/in the hospital as an institution), but for a physical room, i is the normal choice.

Why does it say et—what does that tell me about rom?

et is the indefinite article for neuter nouns.

et rom = a room (neuter)
If it were a common gender noun, you’d expect en (e.g., en stol).

Why is the adjective stille the same with et rom—shouldn’t neuter be different?

Many adjectives have a special neuter ending -t (e.g., en fin bil / et fint hus).
But adjectives ending in -e like stille usually don’t add -t in neuter:

en stille person / et stille rom

If it were definite, you’d typically get the weak form (still stille):

det stille rommet = the quiet room

What’s the basic word order in this sentence?

It’s standard main-clause Norwegian word order (similar to English here):

Subject: Sykepleieren
Verb: ber
Object: de pårørende
Infinitive complement: vente
Place phrase: i et stille rom

If you moved the place phrase to the front, Norwegian would use V2 word order (verb still second):

I et stille rom ber sykepleieren de pårørende vente.