Ideen høres vanvittig ut.

Because ideen means the idea, while en idé means an idea.

Norwegian often puts definiteness on the end of the noun:

• en idé = an idea
• ideen = the idea

So the sentence is talking about a specific idea, not just any idea.

You may also sometimes see the spelling idéen. That is the same word in the definite form.

In Norwegian, the definite article is usually attached to the noun as a suffix instead of standing alone in front.

So:

• English: the idea
• Norwegian: ideen

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

• ideen = the idea
• den vanvittige ideen = the crazy idea

Høres ... ut means sounds or seems, especially based on how something is described.

So Ideen høres vanvittig ut means the idea gives the impression of being crazy.

The word ut does not mean literal out here. It is part of the expression å høres ut, which works a bit like a phrasal verb.

Because å høre and å høres are not used the same way.

• å høre = to hear
• å høres ut = to sound

Examples:

• Jeg hører musikk. = I hear music.
• Det høres bra ut. = That sounds good.

So in this sentence, høres is the correct form because the meaning is sounds, not hears.

Because å høres ut is a separable expression.

In a normal main clause, the finite verb comes early, but the particle ut stays later in the sentence:

• Ideen høres vanvittig ut.

This is very common in Norwegian.

You can think of it like this:

• verb part: høres
• particle: ut

Together they make the meaning sounds / seems.

Norwegian main clauses usually follow the verb-second rule. That means the finite verb comes in the second position.

Here:

1. Ideen = subject
2. høres = finite verb
3. vanvittig = description
4. ut = particle

So the structure is completely normal Norwegian word order.

If you made it a yes/no question, the verb would come first:

• Høres ideen vanvittig ut? = Does the idea sound crazy?

Yes, this is the normal adjective form here.

Vanvittig is a predicate adjective, meaning it comes after a verb like er, virker, or høres ut.

With ideen (a singular noun), vanvittig is the expected form:

• Ideen er vanvittig.
• Ideen høres vanvittig ut.

In the plural, you would normally get:

• Ideene høres vanvittige ut. = The ideas sound crazy.

Also, adjectives ending in -ig often keep the same form in neuter singular too, so vanvittig is a very stable-looking form.

Not always.

Literally, vanvittig means something like crazy, insane, or wild. Depending on context, it can sound:

• negative: That’s a crazy idea
• impressed: That’s a wild/amazing idea
• playful or dramatic

It can also be used as an intensifier:

• vanvittig bra = insanely good
• vanvittig dyrt = insanely expensive

But in Ideen høres vanvittig ut, it describes the idea itself.

Yes, but the nuance changes a little.

• Ideen høres vanvittig ut = The idea sounds crazy
  This focuses on how it comes across when you hear about it.

• Ideen virker vanvittig = The idea seems crazy
  This is more general and not tied as strongly to sound or wording.

You may also hear låter in some varieties of Norwegian:

• Ideen låter vanvittig

But høres ... ut is the safest standard choice for learners in this kind of sentence.

You usually put ikke before the description, while ut stays at the end:

• Ideen høres ikke vanvittig ut. = The idea doesn’t sound crazy.

That same pattern appears in many similar sentences:

• Det høres ikke bra ut. = That doesn’t sound good.
• Planen høres ikke dum ut. = The plan doesn’t sound stupid.