Bogen vil blive lagt på bordet i stuen.

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

• bog = book
• bogen = the book

So -en here is the definite ending. Danish often does this instead of using a separate word like English the.

Because Danish nouns have two grammatical genders:

• common gender nouns usually take -en in the definite singular
• neuter nouns usually take -et in the definite singular

So:

• bog → bogen = the book
• stue → stuen = the living room / the room
• bord → bordet = the table

This is why the endings are different.

This is a future passive construction.

Breakdown:

• vil = will
• blive = be / become
• lagt = past participle of lægge (put / lay)

Together, vil blive lagt means will be placed / will be put.

So the sentence structure is basically:

• Bogen = the book
• vil blive lagt = will be placed
• på bordet i stuen = on the table in the living room

Because the sentence focuses on the book, not on the person doing the action.

Passive means the thing receiving the action becomes the subject:

• Active: Nogen vil lægge bogen på bordet i stuen.
  = Someone will put the book on the table in the living room.

• Passive: Bogen vil blive lagt på bordet i stuen.
  = The book will be put on the table in the living room.

This is very similar to English.

Because after blive in this passive construction, Danish uses the past participle.

• infinitive: lægge = to put / to lay
• past participle: lagt = put / laid

So:

• vil lægge = will put
• vil blive lagt = will be put

Because the two prepositions express different kinds of location:

• på = on
• i = in

So:

• på bordet = on the table
• i stuen = in the living room

This is very close to English usage in this sentence.

Danish often puts the more specific location first, followed by a larger location that contains it.

So:

• på bordet = on the table
• i stuen = in the living room

Together: on the table in the living room

This is also natural in English. The table is located in the living room, so i stuen tells you which table/location is meant.

Yes. That would usually mean The book is being placed / is placed on the table in the living room, depending on context.

The difference is:

• vil blive lagt = will be placed → future
• bliver lagt = is being placed / is placed → present

So vil is what makes the sentence clearly future.

Yes. Danish often expresses future in more than one way.

For example:

• Bogen bliver lagt på bordet i stuen.
• Bogen kommer til at blive lagt på bordet i stuen.
• Bogen vil blive lagt på bordet i stuen.

These can all refer to the future, but the nuance changes:

• vil often sounds like a neutral future, or sometimes implies intention/prediction
• kommer til at often sounds like is going to
• present tense can also refer to the future if the context is clear

A natural active version would be:

Nogen vil lægge bogen på bordet i stuen.

That means Someone will put the book on the table in the living room.

You could also use a specific subject:

• Han vil lægge bogen på bordet i stuen. = He will put the book on the table in the living room.
• Hun vil lægge bogen på bordet i stuen. = She will put the book on the table in the living room.

Because in Danish, definiteness is usually shown by an ending on the noun instead of a separate article.

So:

• bog = a book

• bogen = the book

• bord = a table

• bordet = the table

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

English uses a separate word (the), but Danish normally adds the definite ending directly to the noun.

Very often, yes, but not always in a strictly literal sense.

Stue commonly refers to a living room, and stuen often means the living room. In some contexts it can more broadly mean the main room/parlor/lounge area, depending on style and situation.

In everyday modern Danish, though, translating i stuen as in the living room is usually a very good choice.