Der Schwanz der Katze bewegt sich langsam, wenn sie schläfrig ist.

They are doing two different jobs:

• Der Schwanz: here der is the nominative masculine singular article, because Schwanz is the subject of the sentence.
• der Katze: here der is the genitive feminine singular article, showing possession: the cat’s.

So Der Schwanz der Katze literally means the tail of the cat, or more naturally in English, the cat’s tail.

Because Katze is in the genitive case here.

German often uses the genitive to show possession:

• der Schwanz der Katze = the cat’s tail
• literally: the tail of the cat

The base form is:

• die Katze = the cat

But in the genitive singular, feminine nouns use:

• der Katze

So the noun itself stays Katze, but the article changes from die to der.

Yes, you could, and native speakers do sometimes use von phrases in everyday speech:

• der Schwanz von der Katze

But der Schwanz der Katze is more standard and more elegant in careful written German.

So both can be understood, but the sentence you were given uses the more textbook-style genitive structure.

Sich bewegen means to move.

In German, many verbs are used with a reflexive pronoun, even when English does not use one. So:

• Der Schwanz bewegt sich. = The tail moves / is moving.

You should think of sich bewegen as a unit here.

This does not necessarily mean the tail is deliberately moving itself in a strong English reflexive sense. It is just the normal German way to express this idea.

Because in a normal German main clause, the finite verb usually comes in second position.

Here the structure is:

• Der Schwanz = position 1
• bewegt = finite verb in position 2
• sich = reflexive pronoun
• langsam = adverb

So:

• Der Schwanz bewegt sich langsam.

That word order is completely normal.

Because langsam is being used as an adverb, not as an adjective before a noun.

It describes how the tail moves:

• bewegt sich langsam = moves slowly

Adverbs in German usually do not take adjective endings.

Compare:

• ein langsamer Schwanz = a slow tail
  
  • here langsamer is an adjective before a noun, so it gets an ending
• Der Schwanz bewegt sich langsam = the tail moves slowly
  
  • here langsam is an adverb, so no ending

Sie refers to die Katze.

That is because Katze is a feminine singular noun, and the matching pronoun is sie.

It cannot refer to der Schwanz, because Schwanz is masculine. If it referred to Schwanz, the pronoun would be er.

So:

• die Katze → sie
• der Schwanz → er

In this sentence, wenn sie schläfrig ist means when the cat is sleepy.

Because wenn introduces a subordinate clause, and in German subordinate clauses the conjugated verb usually goes to the end.

So:

• sie ist schläfrig = she is sleepy
  
  • normal main clause word order
• wenn sie schläfrig ist = when she is sleepy
  
  • subordinate clause, so ist moves to the end

This is one of the most important German word-order patterns to learn.

Wenn can mean when, whenever, or if, depending on context.

In this sentence, it most naturally means something like:

• when
• whenever

So the idea is:

• The cat’s tail moves slowly when it is sleepy.

It describes a general situation, not one single event at one exact moment.

Not exactly.

• schläfrig = sleepy, drowsy
• müde = tired

They are related, but not identical. Someone can be:

• müde without necessarily feeling ready to fall asleep
• schläfrig when they feel sleepy or half-asleep

So schläfrig is a good match for English sleepy.

German often avoids repeating possession when the meaning is already clear.

In English, you might say:

• The cat’s tail moves slowly when it is sleepy.

German simply says:

• Der Schwanz der Katze bewegt sich langsam, wenn sie schläfrig ist.

The sie already clearly refers to the cat, so there is no need for an extra possessive word like its.

It is in the present tense, but German present tense can often translate into English in more than one way:

• moves
• is moving

So:

• Der Schwanz der Katze bewegt sich langsam
  can mean
  The cat’s tail moves slowly
  or
  The cat’s tail is moving slowly

The exact English translation depends on context.

Yes. If the subordinate clause comes first, the main clause still keeps the finite verb in second position, which means the subject comes after the verb.

For example:

• Wenn sie schläfrig ist, bewegt sich der Schwanz der Katze langsam.

Notice the main clause is now:

• bewegt sich der Schwanz der Katze langsam

not

• der Schwanz der Katze bewegt sich langsam

This happens because the whole wenn clause takes the first position in the sentence.