Nach dem Waschen hänge ich die Socken mit einer Wäscheklammer auf und stelle das Waschpulver zurück in den Schrank.

Nach is a preposition meaning after, and it always takes the dative case.

So:

• nach + dative
• das Waschen → dem Waschen in dative

That is why the sentence has Nach dem Waschen.

Also, Waschen is being used as a noun here, not as a verb, so it is capitalized. In English, this is similar to after washing or after the washing.

In German, all nouns are capitalized.

Here, waschen is originally a verb, but in dem Waschen it has been turned into a nominalized verb—basically, a verb used as a noun.

So:

• waschen = to wash
• das Waschen = washing

That is why it begins with a capital letter: Waschen.

Because Nach dem Waschen comes first, the sentence follows the normal German verb-second rule.

In a main clause, the finite verb must be in the second position. If something other than the subject comes first, the subject moves after the verb.

So the structure is:

• Nach dem Waschen = first position
• hänge = second position
• ich = then the subject

This is very common in German.

Compare:

• Ich hänge die Socken auf.
• Nach dem Waschen hänge ich die Socken auf.

Because aufhängen is a separable verb.

Its full dictionary form is:

• aufhängen = to hang up

In a main clause, separable verbs split:

• Ich hänge die Socken auf.

So:

• hänge = the conjugated verb part
• auf = the separable prefix, which goes to the end of the clause

This is one of the most important patterns in German.

For the same reason: zurückstellen is also a separable verb.

• zurückstellen = to put back / return something to its place

In a main clause, it splits:

• Ich stelle das Waschpulver zurück.

In your sentence, it becomes:

• ... und stelle das Waschpulver zurück in den Schrank.

So stelle is the conjugated part, and zurück is the separable prefix.

Because the subject is the same in both parts of the sentence: ich.

German often leaves out the repeated subject in coordinated clauses when it is obvious:

• ... hänge ich die Socken auf und stelle das Waschpulver zurück ...

This is similar to English:

• I hang up the socks and put the detergent back in the cupboard.

You could repeat ich, but it is usually unnecessary here.

Because mit always takes the dative case.

The noun is:

• die Wäscheklammer = the clothespin / clothes peg

In dative singular:

• eine Wäscheklammer → einer Wäscheklammer

So:

• mit einer Wäscheklammer = with a clothespin

Because in is a two-way preposition.

That means it can take:

• dative for location: where something is
• accusative for direction/movement: where something is going

Here, the detergent is being moved back into the cupboard, so German uses the accusative:

• in den Schrank

Compare:

• Das Waschpulver ist im Schrank. = location, so dative
• Ich stelle das Waschpulver in den Schrank. = movement into it, so accusative

Because Schrank is a masculine noun:

• der Schrank

After in with movement, you need the accusative. The masculine accusative article changes:

• der → den

So:

• in den Schrank

Socken is the plural of die Socke.

In the plural, the definite article is also die:

• die Socke = the sock
• die Socken = the socks

Here, die Socken is the direct object of hänge ... auf, so it is in the accusative—but plural die stays die in both nominative and accusative.

Because Waschpulver is a neuter noun:

• das Waschpulver

It is the direct object of stelle ... zurück, so it is in the accusative. For neuter nouns, the definite article stays the same in nominative and accusative:

• das → das

So das Waschpulver looks unchanged.

Hängen by itself often refers to hanging or being in a hanging position, while aufhängen means to hang something up.

In this sentence, the idea is not just that the socks are hanging, but that the speaker actively hangs them up. So the separable verb aufhängen is the natural choice.

Very roughly:

• Die Socken hängen. = The socks are hanging.
• Ich hänge die Socken auf. = I hang the socks up.

Because zurück belongs to the separable verb zurückstellen, and in den Schrank adds extra information about destination.

German often places the separable prefix near the end of the clause, but other elements can still follow it, especially prepositional phrases.

So this is normal:

• ... stelle das Waschpulver zurück in den Schrank.

You can think of it as:

• zurückstellen = put back
• in den Schrank = into the cupboard

Yes, that is grammatically correct.

Repeating ich is possible, but it sounds a bit more explicit or heavier. In normal German, if the subject stays the same, it is often omitted after und:

• more natural here: ... und stelle das Waschpulver zurück ...
• more explicit: ... und ich stelle das Waschpulver zurück ...

Both are correct.

They are very similar in meaning, but the structure is different.

• Nach dem Waschen = after washing
• Nachdem ich gewaschen habe = after I have washed / after I washed

The version in your sentence is more compact and uses a noun phrase. The nachdem version uses a subordinate clause with a full verb.

Both are common, but Nach dem Waschen is shorter and very natural here.