Ich hänge die Jacke auf den Kleiderbügel.

Because ich needs the 1st person singular form of the verb.

The infinitive is hängen = to hang.

In the present tense:

• ich hänge
• du hängst
• er/sie/es hängt
• wir hängen
• ihr hängt
• sie/Sie hängen

So Ich hänge ... means I hang ... / I am hanging ....

Die Jacke is the direct object, so it is in the accusative case.

You are doing the action of hanging, and the jacket is the thing being hung.

A useful question is:

• What am I hanging? → die Jacke

Because Jacke is feminine, the article is die in both nominative and accusative singular, so the form does not change here:

• nominative: die Jacke
• accusative: die Jacke

Because auf is a two-way preposition.

Two-way prepositions use:

• accusative for movement toward a destination
• dative for location/static position

Here, the jacket is being moved onto the hanger, so German uses accusative:

• Ich hänge die Jacke auf den Kleiderbügel.
  = I hang the jacket onto the hanger.

But if the jacket is already there, you use dative:

• Die Jacke hängt auf dem Kleiderbügel.
  = The jacket is hanging on the hanger.

So:

• auf den Kleiderbügel = motion/destination
• auf dem Kleiderbügel = location

Here auf means something like onto or on.

With clothes and hangers, German often uses auf, because the clothing is thought of as being placed onto/over the hanger.

So in this sentence:

• auf den Kleiderbügel = onto the hanger

English often just says on the hanger, but German makes the motion/location difference more clearly through case:

• auf den Kleiderbügel = onto the hanger
• auf dem Kleiderbügel = on the hanger

No, not in this sentence.

Here, auf belongs to the prepositional phrase auf den Kleiderbügel.

So the structure is:

• Ich = subject
• hänge = verb
• die Jacke = direct object
• auf den Kleiderbügel = prepositional phrase

That is different from the separable verb aufhängen.

For example:

• Ich hänge die Jacke auf.
  = I hang up the jacket.

In your sentence, auf is not a separated verb prefix; it is simply the preposition introducing den Kleiderbügel.

The difference is action vs. state.

• Ich hänge die Jacke auf den Kleiderbügel.
  Someone is actively putting the jacket onto the hanger.

• Die Jacke hängt auf dem Kleiderbügel.
  The jacket is already there; this describes its position.

So this is a very common German pattern:

• accusative after a two-way preposition = movement to a place
• dative after a two-way preposition = being in a place

Because den means the and refers to a specific hanger.

• auf den Kleiderbügel = onto the hanger
• auf einen Kleiderbügel = onto a hanger

Both are grammatically possible, but they mean different things:

• den = a particular hanger, probably already known from the situation
• einen = any hanger / one hanger

So the sentence with den sounds like the speaker has a specific hanger in mind.

Yes, in some contexts you can, but the nuance is slightly different.

• auf den Kleiderbügel hängen is very common for putting clothing onto a hanger
• an den Kleiderbügel hängen can also be understood, often with a sense of hanging something on/against it

For clothing, auf einen Bügel / auf einen Kleiderbügel hängen is especially natural, because the garment is placed onto the hanger.

So for a learner, auf den Kleiderbügel is a good and useful pattern to remember.

Kleiderbügel is a compound noun:

• Kleider = clothes / garments
• Bügel = hanger, frame, bar

So Kleiderbügel literally means something like clothes hanger.

Its gender is masculine because the last part, Bügel, is masculine:

• der Kleiderbügel = the hanger
• den Kleiderbügel = accusative singular
• dem Kleiderbügel = dative singular

Plural:

• die Kleiderbügel

Because many feminine nouns in German do not change their noun form in the accusative singular.

Compare:

• nominative: die Jacke
• accusative: die Jacke

So the case is shown mainly by the sentence structure and the article, not by a visible change in the noun itself.

This is different from masculine nouns, where the article often changes clearly:

• nominative: der Kleiderbügel
• accusative: den Kleiderbügel

That is why learners often notice the case change more easily on den Kleiderbügel than on die Jacke.

Yes, it is the same basic verb hängen, but German often makes a distinction between:

• transitive: hanging something
• intransitive: something is hanging

In the present tense, the forms look very similar:

• Ich hänge die Jacke ... = I hang the jacket ...
• Die Jacke hängt ... = The jacket hangs / is hanging ...

But in past forms, German often distinguishes them:

Transitive: someone hangs something

• Ich hängte die Jacke auf.
• Ich habe die Jacke aufgehängt.

Intransitive: something is hanging

• Die Jacke hing am Haken.
• Die Jacke hat dort gehangen.

In everyday speech, usage can vary, but this distinction is very useful to know.