Der Mantel bleibt am Haken hängen.

Because der Mantel is the subject of the sentence, so it must be in the nominative case.

The verb bleiben does not take a direct object here. The coat is the thing that stays or gets caught, so German uses nominative:

• der Mantel = the subject

Compare:

• Der Mantel bleibt am Haken hängen.
• Ich hänge den Mantel an den Haken.

In the second sentence, den Mantel is accusative because it is the direct object of hängen.

am is a contraction of an dem.

So:

• am Haken = an dem Haken

This is very common in German:

• am = an dem
• im = in dem

So am Haken is just the normal, shorter form.

Yes, you can, and it is grammatically correct:

• Der Mantel bleibt an dem Haken hängen.

But in normal everyday German, am Haken sounds much more natural. The full form an dem Haken is usually only used when you want extra emphasis or clarity.

Because an is a two-way preposition. That means it can take:

• dative for location: where something is
• accusative for direction/change of position: where something is going

Here the coat is understood as being at/on the hook, so German uses dative:

• am Haken = an dem Haken

Compare:

• Der Mantel hängt am Haken. = location
• Ich hänge den Mantel an den Haken. = movement toward the hook

So the contrast is basically:

• wo? → dative
• wohin? → accusative

Because this sentence uses a verb combination: bleiben + hängen.

In German main clauses, the finite verb goes in second position, and the other verb usually goes to the end:

• Der Mantel bleibt am Haken hängen.

Here:

• bleibt = finite verb
• hängen = second verb, placed at the end

This is very common in German verb structures.

You can compare it with:

• Er bleibt stehen.
• Sie bleibt sitzen.

Because after bleiben in expressions like hängen bleiben, stehen bleiben, liegen bleiben, German uses a bare infinitive.

So you say:

• bleibt hängen
• bleibt stehen
• bleibt liegen

not:

• bleibt zu hängen

This is just how this kind of verb combination works.

This is a very useful distinction.

• Der Mantel hängt am Haken.
  This simply describes the state: the coat is hanging on the hook.

• Der Mantel bleibt am Haken hängen.
  This often suggests that the coat gets caught on the hook or ends up remaining stuck/hanging there.

So hängen alone often describes a static situation, while hängen bleiben often adds the idea of getting stuck or remaining caught.

Here it is intransitive.

That means no direct object is being acted on. The coat is not something that someone is hanging; it is the thing that is hanging / getting caught.

Compare:

• Der Mantel hängt am Haken.
  Intransitive: the coat is hanging.

• Ich hänge den Mantel an den Haken.
  Transitive: I hang the coat on the hook.

This verb is tricky for learners because German uses the same basic form hängen in both patterns, but the grammar is different.

Yes, it is a very common fixed expression in German.

hängen bleiben often means:

• to get caught
• to get stuck
• to remain hanging

So in this sentence, it is best to understand bleibt ... hängen as one unit of meaning, not as two completely separate ideas.

That is why learners often memorize expressions like these as chunks:

• stehen bleiben
• liegen bleiben
• sitzen bleiben
• hängen bleiben

The sentence can change word order, because German is more flexible than English, but the standard neutral order is:

• Der Mantel bleibt am Haken hängen.

You could also say:

• Am Haken bleibt der Mantel hängen.

That version puts more emphasis on am Haken.

What usually stays the same in a main clause is:

• the finite verb bleibt in second position
• hängen at the end

So the middle parts can move around more than they can in English.