Pälsen växer i takt med att katten återhämtar sig.

Päls means fur or coat/fur coat as a noun.
Pälsen is the definite singular form, meaning the fur.

Swedish usually adds the definite article to the end of the noun:

• en päls = a fur / a coat
• pälsen = the fur / the coat

In this sentence, it refers to a specific fur: the cat’s fur.

Katten is the definite singular form of katt, so it means the cat.

• en katt = a cat
• katten = the cat

The sentence is talking about a particular cat, not just any cat. Swedish uses the definite form here for the same reason English says the cat.

I takt med att is a common expression meaning something like:

• as
• in step with
• at the same rate as

So it shows that two processes happen alongside each other.

In this sentence, the idea is that the fur grows gradually while the cat recovers.

You can think of the structure like this:

• i takt med
  • noun = in step with
• i takt med att
  • clause = as / in step with the fact that

For example:

• i takt med tiden = with time / as time passes
• i takt med att katten återhämtar sig = as the cat recovers

Växer is the present tense of växa, which means to grow.

• växa = to grow
• växer = grows / is growing

In Swedish, the present tense does not change depending on the subject:

• jag växer
• du växer
• katten växer

So pälsen växer simply means the fur grows or the fur is growing, depending on context.

Återhämta sig is a reflexive verb, and it means to recover.

The word sig is the reflexive pronoun used for he/she/it/they. Here it refers back to katten.

So:

• katten återhämtar sig = the cat is recovering

This is similar to reflexive verbs in some other European languages, though English usually does not use itself here.

Important point: you normally learn this verb as a whole expression:

• att återhämta sig = to recover

So the sig is not optional in normal usage.

After att here, you have a subordinate clause:

• att katten återhämtar sig

In that clause, the normal order is:

subject + verb + reflexive pronoun

So:

• katten = subject
• återhämtar = verb
• sig = reflexive pronoun

This is different from Swedish main-clause word order, where the verb often has to be in the second position.

The full sentence has:

• main clause: Pälsen växer
• subordinate clause: att katten återhämtar sig

Sometimes yes, but the meaning changes slightly.

• när katten återhämtar sig = when/as the cat recovers
• i takt med att katten återhämtar sig = as the cat recovers, step by step / progressively

I takt med att emphasizes that the two developments happen gradually together. It suggests a parallel process.

So if you want the idea of the fur growing more and more as recovery progresses, i takt med att is a very good choice.

Yes. You would usually change both verbs to past tense:

• Pälsen växte i takt med att katten återhämtade sig.

That means The fur grew as the cat recovered.

The verb changes are:

• växer → växte
• återhämtar sig → återhämtade sig

So the structure stays the same; only the tense changes.

Not exactly.

• hår = hair
• päls = fur or an animal’s coat

For a cat, päls is the natural word if you mean its fur as a coat covering the body.
If you said hår, it would sound more like individual hairs rather than the animal’s fur as a whole.

So pälsen is the most natural choice here.