Die nassen Socken hänge ich mit einer Wäscheklammer auf den Balkon.

Because German main clauses follow the verb-second rule: the finite verb must be in the second position.

Here, Die nassen Socken is placed first for emphasis or topic, so the finite verb hänge comes second, and the subject ich comes after it:

• Die nassen Socken | hänge | ich ...

A more neutral version would be:

• Ich hänge die nassen Socken mit einer Wäscheklammer auf den Balkon auf.

Both are grammatical, but the original sentence highlights the wet socks.

It is one verb: aufhängen, which means to hang up.

German has many separable verbs. In a main clause, the prefix auf- splits off and goes to the end:

• Ich hänge die Socken auf.

That is why you see:

• hänge ... auf

But in other structures, the verb stays together:

• infinitive: die Socken aufzuhängen
• subordinate clause: ..., weil ich die Socken aufhänge.

German is much more flexible than English about what comes first in the sentence.

Putting Die nassen Socken first makes them the topic or gives them extra emphasis. It can sound like:

• As for the wet socks, I’m hanging them...
• The wet socks are what I’m hanging...

This is very common in German and does not sound unnatural.

It is the accusative plural, because it is the direct object of aufhängen: these are the things being hung up.

A confusing point is that in the plural, nominative and accusative often look the same:

• nominative plural: die nassen Socken
• accusative plural: die nassen Socken

So even though the form looks the same, the function here is accusative.

Because of adjective endings.

The noun is plural and has the definite article die, so the adjective takes the ending -en:

• die nassen Socken

This is the normal pattern for adjectives after a definite article in the plural.

Compare:

• die nassen Socken
• mit nassen Socken
• nasse Socken

The ending changes depending on the article and the case.

Because mit always takes the dative case.

The noun is die Wäscheklammer, which is feminine. In the dative singular, the article becomes einer:

• nominative: die Wäscheklammer
• dative after mit: mit einer Wäscheklammer

So mit einer Wäscheklammer means with a clothespin / clothes peg.

Because auf is a two-way preposition. It can take:

• accusative for direction / movement toward a destination
• dative for location

So:

• auf den Balkon = onto / out onto the balcony
• auf dem Balkon = on the balcony

In this sentence, there is an idea of placing the socks somewhere, so German uses the accusative.

Compare:

• Ich hänge die Socken auf den Balkon auf.
  I hang the socks out onto the balcony.

• Die Socken hängen auf dem Balkon.
  The socks are hanging on the balcony.

Yes, literally it does, but idiomatically it can mean something like out on the balcony or up on the balcony area/railing.

German often uses this kind of directional phrase where English might phrase it differently. In context, it means the socks are being hung outside on the balcony, not necessarily placed on top of the balcony floor in a strange way.

So the German is normal, even if the most literal English wording sounds odd.

Wäscheklammer is a compound noun:

• Wäsche = laundry
• Klammer = clip / clamp

So Wäscheklammer means clothespin or clothes peg.

It is feminine:

• die Wäscheklammer

Yes. German word order is flexible, as long as the main rules are respected, especially verb-second in main clauses and the separated prefix auf at the end.

For example:

• Ich hänge die nassen Socken mit einer Wäscheklammer auf den Balkon auf.
• Mit einer Wäscheklammer hänge ich die nassen Socken auf den Balkon auf.
• Auf den Balkon hänge ich die nassen Socken mit einer Wäscheklammer auf.

These versions shift the emphasis:

• starting with Ich = more neutral
• starting with Mit einer Wäscheklammer = emphasizes the tool
• starting with Auf den Balkon = emphasizes the destination

In this sentence, yes: the verb behaves like the transitive weak verb aufhängen.

That means:

• present: ich hänge
• simple past: ich hängte
• perfect: ich habe aufgehängt

This is different from the intransitive sense meaning to be hanging, which often has different past forms:

• Die Socken hängen am Balkon.
• past: Die Socken hingen am Balkon.

So in your sentence, because someone is actively hanging something up, the verb is the transitive one: aufhängen.