Jeg vil hænge billedet op i stuen.

It can be a bit of both.

In Danish, vil often expresses:

• wish / desire: want to
• intention / willingness: am going to, intend to
• sometimes simple future, depending on context

So Jeg vil hænge billedet op i stuen can feel like:

• I want to hang the picture up in the living room
• I’m going to hang the picture up in the living room

A native speaker often hears a nuance of intention or willingness, not just a neutral future.

Because vil is a modal verb, and after modal verbs Danish uses the bare infinitive.

So you say:

• Jeg vil hænge billedet op
• Jeg kan hænge billedet op
• Jeg skal hænge billedet op

not:

• Jeg vil at hænge...

This is similar to English:

• I will hang
• I can hang

not:

• I will to hang

For the same reason: after vil, the main verb stays in the infinitive.

• vil hænge = want to hang / will hang
• hænger = hangs / am hanging in present tense

Compare:

• Jeg vil hænge billedet op = I want to / will hang the picture up
• Jeg hænger billedet op = I’m hanging the picture up / I hang the picture up

So hænge is correct because it depends on vil.

Billedet means the picture.

The basic noun is:

• et billede = a picture

When it becomes definite, Danish usually adds the definite article to the end of the noun:

• billedet = the picture

So:

• et billede = a picture
• billedet = the picture

The ending -et shows that billede is a neuter noun (et-word).

Because Danish usually puts the definite article on the noun itself.

Examples:

• et billede → billedet
• en stue → stuen

So instead of saying a separate word like English the picture, Danish often says picture-the, so to speak.

A separate definite word appears especially when there is an adjective:

• billedet = the picture
• det store billede = the big picture

and:

• stuen = the living room
• den store stue = the big living room

Op means up, and in this sentence it is an important part of the expression hænge ... op.

Hænge op means:

• hang up
• put up
• especially attach something so it hangs, often on a wall

So:

• hænge billedet op = hang the picture up / put the picture up

Without op, the meaning becomes less specific. Hænge by itself can mean hang more generally.

So op is very natural here and is normally what you want if you mean putting a picture on the wall.

Because Danish often places the object between the verb and the particle in expressions like this.

So the normal pattern is:

• hænge billedet op
• sætte lampen op
• skrive nummeret ned

This works much like English hang the picture up.

With a pronoun, Danish also normally keeps that order:

• hænge det op = hang it up

So even though hænge op belongs together in meaning, the object often comes in the middle.

I stuen means in the living room.

Breakdown:

• i = in
• stuen = the living room

A useful point: stue usually means living room / sitting room / lounge, not just any room.
A general room is more often værelse.

So:

• i stuen = in the living room
• i værelset = in the room / in the bedroom depending on context

Because i is the normal preposition for being inside a room.

So:

• i stuen = in the living room
• i køkkenet = in the kitchen
• i soveværelset = in the bedroom

På is used for other kinds of places, but for ordinary rooms in a house, i is the standard choice.

Yes. Danish hænge can be both:

1. transitive: someone hangs something

   • Jeg hænger billedet op = I hang / am hanging the picture up

2. intransitive: something is hanging

   • Billedet hænger på væggen = The picture is hanging on the wall

English often separates these ideas a bit more clearly in everyday use, so this can feel unusual at first. In Danish, context tells you which meaning is intended.

Yes, you could, but the nuance changes.

• Jeg vil hænge billedet op i stuen
  = I want to / intend to hang the picture up in the living room

• Jeg skal hænge billedet op i stuen
  = I’m going to / I have to hang the picture up in the living room

Skal often suggests:

• a plan
• an obligation
• something expected or decided

Vil focuses more on will, desire, or intention.

You would use the indefinite form:

• Jeg vil hænge et billede op i stuen = I want to hang a picture up in the living room

Compare:

• et billede = a picture
• billedet = the picture

So the sentence with billedet refers to a specific picture that both speaker and listener can identify, while et billede introduces a non-specific one.