Lyset i køkkenet er slukket.

Because Danish usually marks the by adding it to the end of the noun.

• et lys = a light
• lyset = the light

So lyset is the definite singular form of lys.

Also, lys is a neuter noun, which is why the indefinite article is et and the definite ending is -et.

For the same reason:

• et køkken = a kitchen
• køkkenet = the kitchen

In the phrase i køkkenet, Danish is saying in the kitchen.

In Danish, the definite article is usually attached to the noun instead of standing in front of it.

So English:

• the light
• the kitchen

becomes Danish:

• lyset
• køkkenet

A separate definite word such as det is normally used when there is an adjective before the noun:

• det store køkken = the big kitchen

But in your sentence there is no adjective before lys or køkken, so the ending alone is enough.

i means in here.

So:

• i køkkenet = in the kitchen

It is a very common preposition and often works like English in.

This is the normal word order for a simple Danish statement:

• Lyset i køkkenet = the subject
• er = the verb
• slukket = the complement describing the subject

So the structure is basically:

[subject] + [verb] + [description]

Danish also follows the verb-second rule in main clauses. In this sentence, the subject comes first, so the verb er comes second.

You could also front another element and still keep the verb second:

• I køkkenet er lyset slukket.

That is also grammatical, but it shifts the emphasis.

Slukket is the past participle of at slukke (to switch off / turn off), but in this sentence it functions like an adjective meaning:

• off
• switched off
• turned off

So er slukket describes a state: the light is in the off state.

No. Er slukket normally describes a resulting state, not an ongoing action.

• Lyset er slukket = the light is off
• Lyset bliver slukket = the light is being turned off

That distinction is very useful in Danish:

• er + participle/adjective often describes a state
• bliver + participle often describes an action/process

It can mean the light in a practical everyday sense, often referring to the lamp or light source as English does in sentences like The kitchen light is off.

So in context, lyset i køkkenet usually means something like:

• the kitchen light
• the light in the kitchen

It does not have to mean light as an abstract substance.

Yes, that can work, but the nuance is slightly different.

• Lyset i køkkenet er slukket focuses on which light: the one in the kitchen.
• Lyset er slukket i køkkenet can sound more like the whole situation is located in the kitchen: In the kitchen, the light is off.

In many everyday situations, both are understandable. The original sentence is the clearest way to say the light in the kitchen.

These are two sounds that English speakers often find tricky.

• y in lyset has no exact English equivalent. It is similar to German ü or French u.
  A good starting point: say ee while rounding your lips.

• ø in køkkenet also has no exact English equivalent. It is similar to German ö or French eu.

A few extra pronunciation notes:

• lyset is roughly two syllables: LY-se(t), with the last part weak.
• køkkenet is roughly KØK-ke-net, but the unstressed endings are reduced in normal speech.
• slukket also has a weak ending in everyday pronunciation.

You do not need perfect pronunciation immediately; getting y and ø approximately right is already a big step.

Because Danish does not capitalize all nouns.

So:

• lyset
• køkkenet

stay lowercase unless they begin the sentence.

The only reason Lyset has a capital letter here is that it is the first word of the sentence.