I takt med att hunden återhämtade sig ökade aptiten.

It is a common linking expression meaning something like as, in step with, or at the same rate as. It shows that two changes happen gradually together.

So here, the dog's recovery and the increase in appetite develop side by side over time.

Yes, it is best to learn it as a set phrase.

• i takt med can mean in step with
• i takt med att is the full expression used before a clause

So in this sentence, i takt med att functions as one unit introducing hunden återhämtade sig.

Here att is not the infinitive marker meaning to. It is part of the conjunction i takt med att, which introduces a clause.

Compare:

• i takt med att hunden återhämtade sig = followed by a clause
• i takt med hundens återhämtning = followed by a noun phrase, so no att

No, not if you keep the full clause hunden återhämtade sig.

You can say:

• i takt med att hunden återhämtade sig

But not:

• i takt med hunden återhämtade sig

If you want to drop att, you have to change the structure, for example:

• i takt med hundens återhämtning

Because återhämta sig is the usual Swedish verb for recover in the sense of get better.

It is a reflexive verb, so it needs a reflexive pronoun:

• jag återhämtade mig
• du återhämtade dig
• hunden återhämtade sig

Without sig, återhämta usually means something more like recover or retrieve something.

Both are in the Swedish preterite, which is the simple past.

• återhämtade sig = recovered / was recovering
• ökade = increased / was increasing

Swedish often uses the simple past where English might use either simple past or past progressive, depending on context. This sentence describes an ongoing development in the past, so English may translate it in different natural ways.

This is because Swedish main clauses follow the V2 rule: the finite verb normally comes in the second position.

The sentence begins with the fronted clause:

I takt med att hunden återhämtade sig

That whole clause takes the first position. So the finite verb of the main clause, ökade, must come next. After that comes the subject aptiten.

That is why you get:

• I takt med att hunden återhämtade sig ökade aptiten

instead of:

• I takt med att hunden återhämtade sig aptiten ökade

The basic main-clause order is:

Aptiten ökade.

When the longer time/process clause is added at the end, you get:

Aptiten ökade i takt med att hunden återhämtade sig.

So the version in your sentence is just the same idea with the subordinate clause moved to the front for emphasis.

There are two clauses, so there are two subjects:

• In hunden återhämtade sig, the subject is hunden
• In ökade aptiten, the subject is aptiten

So the sentence is structurally:

• subordinate clause: the dog recovered
• main clause: the appetite increased

Because Swedish is referring to specific, context-known things.

• hunden = the dog
• aptiten = the appetite

In English, you might naturally say its appetite, but Swedish often uses the definite form when it is already obvious whose appetite is meant. If needed, Swedish could be more explicit with hundens aptit, but that is not necessary here.

Yes, absolutely. That is a very natural sentence too.

The meaning stays the same, but the emphasis changes a little:

• I takt med att hunden återhämtade sig ökade aptiten focuses first on the process or development
• Aptiten ökade i takt med att hunden återhämtade sig starts with aptiten as the topic

I takt med att suggests a gradual, parallel development.

När usually just means when and places something in time.

So:

• När hunden återhämtade sig ökade aptiten can sound more like when the dog recovered, the appetite increased
• I takt med att hunden återhämtade sig ökade aptiten clearly suggests that the appetite increased progressively as the recovery progressed

That gradual side-by-side meaning is why i takt med att is especially suitable here.