Nach dem Spaziergang wasche ich der Hündin die Pfoten, weil sie schmutzig sind.

Because nach is a preposition that normally takes the dative case.

• der Spaziergang = nominative
• den Spaziergang = accusative
• dem Spaziergang = dative

So nach dem Spaziergang means after the walk.

Also, nach is one of the common prepositions that learners memorize with dative, especially in time expressions like this one.

German often puts a time expression first to set the scene: After the walk...

When something other than the subject comes first in a German main clause, the conjugated verb still has to stay in second position. That is why you get:

• Nach dem Spaziergang = first position
• wasche = second position
• ich = after the verb

So:

• Ich wasche der Hündin die Pfoten.
• Nach dem Spaziergang wasche ich der Hündin die Pfoten.

Both are correct, but the second one emphasizes when it happens.

Because der Hündin is dative, not nominative or accusative.

The verb pattern here is:

• jemandem etwas waschen = to wash something for someone / to wash someone’s something

So in this sentence:

• der Hündin = the dog, in the dative
• die Pfoten = the paws, in the accusative

German often uses this structure with body parts and clothing:

• Ich wasche dem Kind die Hände. = I wash the child’s hands.
• Er putzt sich die Zähne. = He brushes his teeth.

So der Hündin die Pfoten waschen is a very natural German way to say wash the dog’s paws.

Because German often prefers a dative possessor structure with body parts.

Instead of saying:

• I wash her paws

German very often says the equivalent of:

• I wash the dog the paws

That sounds odd in English, but it is very normal in German.

So:

• Ich wasche der Hündin die Pfoten.

is more idiomatic than always using a possessive like ihre Pfoten.

Using ihre Pfoten is possible in some contexts, but the dative structure is especially common when the possessor is affected by the action.

Because Pfoten is plural, and here it is the accusative direct object.

The noun is:

• singular: die Pfote
• plural: die Pfoten

In the plural, the nominative and accusative article is often the same: die.

So:

• die Pfoten = the paws

That is the thing being washed, so it is the direct object.

Hündin means female dog.

Compare:

• der Hund = male dog or dog in a general sense
• die Hündin = female dog

So the sentence specifically refers to a female dog.

Its case form here is dative singular:

• nominative: die Hündin
• dative: der Hündin

Because the sentence contains both a main clause and a subordinate clause.

Main clause:

• Nach dem Spaziergang wasche ich der Hündin die Pfoten

In a German main clause, the conjugated verb is in second position.

Subordinate clause:

• weil sie schmutzig sind

A clause introduced by weil is a subordinate clause, and in subordinate clauses the conjugated verb usually goes to the end.

So:

• ..., weil sie schmutzig sind.

That is one of the most important word-order patterns in German.

Here sie refers to die Pfoten.

Why?

• die Pfoten is plural
• sie can mean they
• the verb is sind, which is plural

So the meaning is:

• because they are dirty

meaning because the paws are dirty.

It does not mean because she is dirty, because that would require:

• weil sie schmutzig ist

with ist in the singular.

Because the subject of the weil clause is plural: sie = they, referring to die Pfoten.

Compare:

• die Pfote ... ist schmutzig = the paw is dirty
• die Pfoten ... sind schmutzig = the paws are dirty

So sind tells you that the thing being described as dirty is plural.

Yes, schmutzig is an adjective, but here it is being used predicatively, after the verb sein.

German adjectives behave differently depending on where they appear:

• die schmutzigen Pfoten = attributive adjective, so it takes an ending
• Die Pfoten sind schmutzig = predicative adjective, so it has no ending

In your sentence, it is in the second pattern:

• weil sie schmutzig sind

So schmutzig stays unchanged.

Yes, absolutely.

That version starts with the subordinate clause:

• Weil sie schmutzig sind, wasche ich der Hündin die Pfoten.

This is also correct German. When the subordinate clause comes first, the main clause still follows the usual rule that the conjugated verb comes first within the remaining main-clause part:

• ..., wasche ich ...

Both versions are natural:

• Nach dem Spaziergang wasche ich der Hündin die Pfoten, weil sie schmutzig sind.
• Weil sie schmutzig sind, wasche ich der Hündin die Pfoten.

The difference is mainly one of emphasis.

The main clause is built like this:

• Nach dem Spaziergang = time expression
• wasche = conjugated verb
• ich = subject
• der Hündin = indirect object, dative
• die Pfoten = direct object, accusative

So the core pattern is:

• Ich wasche der Hündin die Pfoten.

And then the time phrase is placed first:

• Nach dem Spaziergang wasche ich der Hündin die Pfoten.

This is a very common German structure: time phrase + verb + subject + objects.