Learn Hardcore Korean: chimsil jomyeongi kkeojyeo isseoyo.

It marks a resultant state. With intransitive/passive verbs like 꺼지다/켜지다, V-아/어 있다 means the subject is in that state as a result of a prior action. So 꺼져 있어요 means it is currently off and remains off.

How is 꺼져 있어요 different from 꺼졌어요?

꺼졌어요: a completed event (it went off/was turned off).
꺼져 있어요: the present state (it is off now). Also, 껐어요 (from 끄다) says someone actively turned it off; 꺼졌어요 doesn’t specify who.

Why is the subject marker -이 used on 조명? Could I use -은 instead?

조명 + 이 marks the grammatical subject of the descriptive predicate 꺼져 있다. 은/는 topicalizes: 침실 조명은 꺼져 있어요 adds contrast (the bedroom light is off, but others might not be). Both are correct; choose based on nuance.

Why not say 침실 조명을 꺼져 있어요?

Because 꺼져 있다 is intransitive/descriptive and cannot take an object. Use:

State: 침실 조명이 꺼져 있어요.
Action you did: 침실 조명을 껐어요. (from 끄다)

Can I use 불, 등, or 전구 instead of 조명? Which is most natural?

불: everyday word for the light (most common): 침실 불이 꺼져 있어요.
등/전등: the lamp/fixture; also natural: 침실 등(이) 꺼져 있어요.
전구: the bulb; use only when you mean the bulb itself.
조명: lighting/illumination; fine, slightly formal/technical.

Is 침실 조명 the same as 침실의 조명 or 침실에 조명?

침실 조명: natural noun–noun compound.
침실의 조명: grammatical but 의 can sound stiff in casual speech.
침실에 조명: locative meaning “lighting in the bedroom.” For this sentence, prefer 침실 조명이… or 침실 불이…. 침실에 조명이 꺼져 있어요 is understandable but clunky.

Does 꺼져 also mean “Go away!” like in dramas? Is that related?

Yes. 꺼져! is a very rude imperative meaning “Get lost!” (from 꺼지다 “to vanish/go out”). In 꺼져 있어요, 꺼져 is just part of the state expression “be off,” unrelated to telling someone to go away.

How would I say it if the light is on instead?

Use 켜지다:

침실 불이 켜져 있어요.
Or, with your noun: 침실 조명이 켜져 있어요.

How do you pronounce the whole sentence naturally?

Roughly: chimsil jomyeong-i kkeojyeo isseoyo.

실 조 often sounds like 실쪼 in fast speech.
있어요 is pronounced 이써요.
In speech it often runs together as 꺼져있어요, though writing with a space is standard.

What politeness level is this? Alternatives?

-어요 is polite informal.

Casual: 꺼져 있어.
Formal: 꺼져 있습니다.
Question: 꺼져 있어요?

Do I need a space in 꺼져 있어요?

Both 꺼져 있어요 and 꺼져있어요 appear, but the standard recommendation is to write auxiliary constructions with a space: 꺼져 있어요. Many people drop the space in texting.

Do I need to make 조명 plural if there are several lights?

No. Korean usually leaves number implicit. 조명이 꺼져 있어요 can refer to one or more lights. Use -들 only for emphasis (e.g., 조명들이 다 꺼져 있어요).

Can I say 침실에서 조명이 꺼져 있어요?

Grammatically yes, but it’s less natural. Prefer 침실 조명이… or 침실 불이…. If you want an explicit location, use 침실에서 불이 꺼져 있어요 or 침실에 있는 조명이 꺼져 있어요.

How would I ask or request something related to this?

Ask: 침실 불이 꺼져 있어요? / 침실 조명이 꺼져 있어요?
Request: 침실 불 좀 꺼 주세요. (Please turn off the bedroom light.)
Report an action: 침실 불을 껐어요.

What’s the difference between 꺼져 있어요 and 꺼져요?

꺼져 있어요: describes the current state (is off now).
꺼져요: can mean “goes off/turns off” (habitual/future) or be ambiguous for a state. For a static state, 꺼져 있어요 is clearer and more natural.