Han vill få jackan lagad, men hon måste få köket städat först.

Here få is part of a very common Swedish pattern:

få + object + past participle

It often means:

• have something done
• get something done

So:

• få jackan lagad = have/get the jacket repaired
• få köket städat = have/get the kitchen cleaned

This is not mainly about permission here. It is about arranging for a result.

Compare:

• Han får gå nu. = He is allowed to go now.
• Han får jackan lagad. = He gets/has the jacket repaired.

Same verb, different construction.

Because the past participle agrees with the noun it describes.

• jackan is a common gender noun (en jacka)
  → participle form: lagad
• köket is a neuter noun (ett kök)
  → participle form: städat

So:

• en jacka → lagad
• ett kök → städat

This is similar to adjective agreement in Swedish.

A few examples:

• en dörr → få dörren målad
• ett hus → få huset målat
• skorna (plural) → få skorna lagade

They are past participles used in an adjective-like way.

They come from verbs:

• laga → lagad
• städa → städad/städat/städade

In this sentence, they describe the state the object should end up in:

• jackan lagad = the jacket repaired
• köket städat = the kitchen cleaned

So they are verb-derived forms, but they behave a lot like adjectives because they agree with the noun.

Because that is the normal structure of this construction:

få + object + past participle

So the object comes before the participle it belongs with.

Pattern:

• få bilen tvättad
• få håret klippt
• få fönstren putsade

The participle describes the object, so they stay together as a unit:

• jackan lagad
• köket städat

That is why få lagad jackan would be wrong.

Because Swedish modal verbs usually take the bare infinitive.

Both vill and måste are modal verbs, so they are followed directly by the infinitive:

• Han vill få ...
• Hon måste få ...

Not:

• Han vill att få ...
• Hon måste att få ...

Other common modal verbs work the same way:

• kan
• ska
• bör
• får
• måste
• vill

Examples:

• Jag kan komma.
• Vi måste gå.
• Hon vill läsa.

The definite forms jackan and köket suggest a specific jacket and a specific kitchen.

So the sentence is talking about:

• the jacket
• the kitchen

not just any jacket or any kitchen.

This is very natural in Swedish when the speaker and listener know which thing is meant.

Compare:

• Han vill få en jacka lagad. = He wants to get a jacket repaired.
• Han vill få jackan lagad. = He wants to get the jacket repaired.

Both are grammatical, but they mean slightly different things.

Not necessarily.

This construction often leaves the agent unspecified. It focuses on the result, not on who performs the action.

So:

• Han vill få jackan lagad does not mean he repairs it himself.
• Hon måste få köket städat does not mean she cleans it herself.

It could mean:

• they do it themselves,
• someone else does it,
• they arrange for it to be done.

The sentence only tells you that they want or need those things to end up repaired/cleaned.

They are related, but not identical.

• städa köket = clean the kitchen
• få köket städat = have/get the kitchen cleaned

The first focuses on the action itself.
The second focuses on achieving the result.

Compare:

• Hon måste städa köket först.
  = She has to clean the kitchen first.
  This strongly suggests she is the one doing it.

• Hon måste få köket städat först.
  = She has to get/have the kitchen cleaned first.
  This focuses more on making sure it gets done.

Först means first, and in this sentence it modifies the whole second clause:

• men hon måste få köket städat först

Putting först at the end is very natural in Swedish. It gives the sense of before anything else or as the first priority.

Swedish adverbs are often fairly flexible, but this final position is especially common and natural here.

Compare:

• Hon måste först få köket städat.
• Hon måste få köket städat först.

Both are possible, but the second one is especially idiomatic in everyday speech.

In practice, no — not in this sentence.

Because of the pattern få + object + past participle, the meaning is clearly about getting something done.

So:

• få jackan lagad = get/have the jacket repaired
• få köket städat = get/have the kitchen cleaned

If få meant be allowed to, you would normally expect a different structure, usually with an infinitive:

• Hon får städa köket. = She is allowed to clean the kitchen.

That is a different grammar pattern and a different meaning.

Yes. It is very productive and useful.

Common examples:

• få bilen lagad = have the car repaired
• få håret klippt = get one’s hair cut
• få fönstren tvättade = have the windows washed
• få brevet skickat = get the letter sent
• få maten levererad = get the food delivered

It is a very handy pattern when you want to talk about arranging for a result rather than focusing on who performs the action.