Stuen er lys, når solen skinner gennem vinduet.

Because stuen is the definite singular form of stue.

In Danish, the definite article is usually attached to the end of the noun:

• en stue = a living room / a room
• stuen = the living room

So Stuen er lys means The living room is bright.

This is one of the big differences from English: Danish often uses an ending instead of a separate word like the.

For the same reason: they are also definite nouns.

• en sol → solen = the sun
• et vindue → vinduet = the window

So Danish is saying:

• solen = the sun
• vinduet = the window

This is very common in Danish. The definite form is often built into the noun itself.

Because stuen is a common-gender singular noun, and predicative adjectives agree with gender and number.

Here lys is a predicative adjective because it comes after er:

• Stuen er lys = The living room is bright

The forms are:

• common gender singular: lys
• neuter singular: lyst
• plural: lyse

Examples:

• Stuen er lys. = The living room is bright.
• Værelset er lyst. = The room is bright.
  (værelse is neuter)
• Rummene er lyse. = The rooms are bright.

So lys is correct because stue is an en-word.

Yes. Lys can be both:

• a noun: light
• an adjective: bright, light-colored

In this sentence, lys is an adjective, because it describes stuen.

So here it means something like:

• bright
• well-lit

It does not mean the noun light here.

Because the sentence describes a state, not a change.

• er = is
• bliver = becomes / gets

So:

• Stuen er lys = The living room is bright
• Stuen bliver lys = The living room becomes / gets bright

In your sentence, the idea is that whenever the sun shines through the window, the room is bright. That is why er fits better.

Når is used for something that happens when / whenever a situation occurs, especially something expected, repeated, or general.

• når = when / whenever
• hvis = if

So:

• Stuen er lys, når solen skinner gennem vinduet.
  = The living room is bright when/whenever the sun shines through the window.

This sounds like a general fact or usual situation.

If you said hvis, it would sound more conditional:

• Stuen er lys, hvis solen skinner gennem vinduet.
  = The living room is bright if the sun shines through the window.

That is not impossible, but når is more natural for a regular, real-world situation like this.

Good question. Danish does change word order, but it depends on the clause structure.

In your sentence, the main clause comes first:

• Stuen er lys = main clause
• når solen skinner gennem vinduet = subordinate clause

So the main clause keeps normal word order:

• subject + verb + complement
• Stuen er lys

Inside the subordinate clause, the word order is also normal for a subordinate clause:

• når solen skinner gennem vinduet

If you put the når-clause first, then the main clause would change:

• Når solen skinner gennem vinduet, er stuen lys.

Notice that now it is er stuen lys, not stuen er lys. That is because Danish uses V2 word order in main clauses.

So both are correct:

• Stuen er lys, når solen skinner gennem vinduet.
• Når solen skinner gennem vinduet, er stuen lys.

Because Danish often uses the present tense for:

• general truths
• habitual situations
• things that happen regularly

Here the sentence describes a normal recurring situation:

• the living room is bright when the sun shines through the window

So Danish uses present tense in both clauses:

• er
• skinner

This works much like English.

Gennem means through.

So:

• gennem vinduet = through the window

It describes the path of the sunlight.

You may also see igennem, which is often very similar. In many contexts, gennem and igennem can both mean through, though gennem is very common and natural here.

Yes, you often can.

• gennem vinduet
• igennem vinduet

Both can mean through the window.

In many everyday contexts, the difference is small or nonexistent. Gennem is usually the simpler, more neutral choice, and it sounds perfectly natural in this sentence.

Usually because Danish marks definiteness with an ending attached to the noun.

Compare:

• English: the room
• Danish: stuen

This is called the postposed definite article.

Very often:

• en/et + noun = indefinite
• noun + -en / -et = definite

Examples:

• en stol = a chair
• stolen = the chair
• et bord = a table
• bordet = the table

Danish can also use a separate definite word, but usually together with an adjective:

• den lyse stue = the bright living room

So Danish has both patterns, but in your sentence there is no adjective before the noun, so the attached ending is the normal choice:

• stuen
• solen
• vinduet

Often, yes, but it depends on context.

In modern everyday Danish, stue very commonly means living room or sitting room. In some contexts it can also refer more generally to a room or to a main room in a house.

In this sentence, living room is the most natural interpretation.

It would become:

• Når solen skinner gennem vinduet, er stuen lys.

This means the same thing, but the sentence starts with the subordinate clause. Because that clause comes first, the main clause must follow Danish V2 word order, so you get:

• er stuen lys not
• stuen er lys

That is a very common pattern in Danish.

Vindue is a neuter noun:

• et vindue = a window

The definite singular is:

• vinduet = the window

So for many neuter nouns, the definite ending is -et.

Compare:

• et hus → huset
• et æble → æblet
• et vindue → vinduet

Because vindue already ends in -e, the result is vindue + t, written vinduet.