Nach dem Spaziergang kühlt meine Mutter den Stich mit kaltem Wasser.

Because nach takes the dative case when it means after in a time expression.

• der Spaziergang = nominative
• dem Spaziergang = dative

So:

• nach dem Spaziergang = after the walk

For this masculine noun, the article changes like this:

• der → nominative
• den → accusative
• dem → dative

That is why dem is correct here.

Because den Stich is the direct object of the verb kühlen.

In this sentence, my mother cools the sting/bite, so the sting is the thing being affected by the action. In German, the direct object usually takes the accusative case.

• der Stich = nominative
• den Stich = accusative

So:

• Meine Mutter = subject
• kühlt = verb
• den Stich = direct object

German main clauses follow the verb-second rule, often called V2.

The first part of the sentence is:

• Nach dem Spaziergang

That whole time expression takes the first position. In a German main clause, the finite verb must then come second:

• Nach dem Spaziergang kühlt meine Mutter den Stich ...

So even though meine Mutter is the subject, it comes after the verb here.

You could also say:

• Meine Mutter kühlt nach dem Spaziergang den Stich mit kaltem Wasser.

That is also grammatical, but it places the emphasis differently.

Yes. Meine Mutter is the subject, so it is in the nominative.

It may not look different because Mutter is feminine, and some feminine forms do not change as visibly as masculine ones.

Here:

• meine Mutter = nominative subject
• den Stich = accusative object

So the sentence structure is basically:

• After the walk, my mother cools the sting with cold water.

Because mit always takes the dative case.

So Wasser must be in the dative, and the adjective must also get a dative ending.

• mit Wasser = with water
• mit kaltem Wasser = with cold water

Why kaltem?

Because:

• Wasser is neuter
• there is no article here
• adjective endings after no article must show the case clearly
• dative singular neuter uses -em

So:

• kaltes Wasser = nominative/accusative
• kaltem Wasser = dative

Stich can mean a sting, bite, prick, or puncture, depending on context.

In a sentence like this, it often refers to something like:

• an insect bite
• an insect sting

The exact English translation depends on the situation. German often uses Stich a bit more broadly than English would use sting.

So if the meaning shown to the learner is bite, that can still fit naturally in many contexts.

Here nach means after, in a temporal sense.

So:

• nach dem Spaziergang = after the walk

This is a very common pattern:

• nach dem Essen = after the meal
• nach der Arbeit = after work
• nach dem Unterricht = after class

Important: nach has several meanings in German, such as to, toward, or after, depending on context. In this sentence, it is clearly the time meaning: after.

Because in German, all nouns are capitalized.

So in this sentence:

• Spaziergang = noun
• Mutter = noun
• Stich = noun
• Wasser = noun

This is one of the most noticeable differences from English. Adjectives and verbs are normally not capitalized unless they begin the sentence.

Yes, but they are not always used in exactly the same way.

• kühlen = to cool something
• abkühlen = to cool down / to let something cool off

In this sentence, kühlt den Stich sounds natural because the mother is applying something cold to the bite/sting in order to relieve it.

Abkühlen can also be used, but kühlen is a straightforward choice for cooling something medically or physically.

Yes, German word order is somewhat flexible, but the finite verb still has to stay in second position in a main clause.

For example, these are possible:

• Nach dem Spaziergang kühlt meine Mutter den Stich mit kaltem Wasser.
• Meine Mutter kühlt nach dem Spaziergang den Stich mit kaltem Wasser.
• Den Stich kühlt meine Mutter nach dem Spaziergang mit kaltem Wasser.

These versions do not all sound equally neutral. The original sentence is a natural, neutral way to begin with the time expression.

So German allows movement for emphasis, but the verb-second rule remains very important.