Für den Salat würfle ich die Paprika klein und hacke danach den Lauch.

Because German main clauses follow the verb-second rule: the conjugated verb must be in the second position, but the first position does not have to be the subject.

So here:

• Für den Salat = first position
• würfle = second position
• ich = comes after the verb

This is a very common German pattern. Starting with Für den Salat puts the focus on the context: as for the salad / for the salad.

A more neutral version would be:

• Ich würfle für den Salat die Paprika klein und hacke danach den Lauch.

But the original sounds very natural.

Because für always takes the accusative case.

The noun is:

• der Salat = nominative

After für, it becomes:

• für den Salat = accusative

So this is not a special feature of this sentence; it is simply the normal case required by für.

Here würfeln is a cooking verb. It means to dice, that is, to cut something into small cubes.

So:

• die Paprika würfeln = to dice the pepper

This is different from the other meaning of würfeln, which can also mean to play dice / roll dice. In a cooking sentence, it clearly means to dice.

Because klein here is not describing the noun directly. It is describing the result of the action.

Compare:

• die kleine Paprika = the small pepper
  
  • Here kleine is an adjective directly modifying Paprika.
• die Paprika klein würfeln = to dice the pepper small / into small pieces

In the sentence, klein works more like a resultative adjective: it tells you how the pepper ends up after being cut.

German often does this with food-preparation verbs:

• klein schneiden = cut small
• fein hacken = chop finely
• dünn schneiden = cut thinly

So klein stays in its basic form.

That comes from the gender of the nouns and the accusative case.

• die Paprika is feminine
• der Lauch is masculine

Both nouns are direct objects here, so they are in the accusative.

In the accusative:

• feminine die stays die
• masculine der changes to den

So:

• die Paprika → die Paprika
• der Lauch → den Lauch

Because German often leaves out a repeated subject in the second part of a coordinated sentence when the subject is the same.

The full version would be:

• Für den Salat würfle ich die Paprika klein und ich hacke danach den Lauch.

But repeating ich is not necessary, so German usually shortens it to:

• ... und hacke danach den Lauch.

This is very common and natural.

Danach means after that or afterwards.

It shows the order of actions:

1. first dice the pepper
2. then chop the leek

Its position is flexible, but the word order must still follow normal German rules.

For example, all of these are possible:

• ... und hacke danach den Lauch.
• ... und danach hacke ich den Lauch.
• ... und ich hacke danach den Lauch.

The original version is compact and natural.

They describe different ways of cutting.

• würfeln = to dice, to cut into cubes
• hacken = to chop, usually into smaller irregular pieces

So in the sentence:

• die Paprika würfeln = cut the pepper into cubes
• den Lauch hacken = chop the leek

This is typical cooking vocabulary, and German often uses quite precise verbs for different cutting methods.