Hon har fått håret klippt, så nu känner hon sig nöjd.

Because Swedish uses the pattern få + object + verb form to mean that someone gets something done.

So:

Hon har klippt håret = she has cut her hair
Hon har fått håret klippt = she has had her hair cut

The second version usually suggests that someone else did the cutting, such as a hairdresser, or at least that the important point is that the haircut happened to her hair.

It often does mean get or receive, but in this kind of sentence it works more like English get/have something done.

So har fått håret klippt is literally close to has got the hair cut, but the natural English meaning is has had her hair cut.

It does not mean permission here, like is allowed to.

With body parts, Swedish very often uses the definite form when it is already clear whose body part it is.

So håret here naturally means her hair because the subject is hon.

This is very common in Swedish:

• Hon tvättade händerna = She washed her hands
• Han bröt benet = He broke his leg

You could say hennes hår in some contexts, but that sounds more emphatic or contrastive. In this sentence, håret is the most natural choice.

After få in this construction, Swedish uses the form seen in patterns like få något gjort.

So:

• klippa = infinitive
• klippte = past tense
• klippt = the form used here

A good way to learn it is as a set pattern:

få håret klippt
få bilen lagad
få fönstret öppnat

It expresses the result: the hair ends up cut.

Here så means so, therefore, or as a result.

It links the two parts of the sentence:

• she has had her hair cut
• therefore she feels pleased now

So så is connecting cause/result.

Because Swedish main clauses follow the verb-second rule.

After så, the next clause begins with nu. Since nu comes first in that clause, the finite verb känner must come second, before the subject hon:

så nu känner hon sig nöjd

This is the normal Swedish pattern:

• Nu känner hon sig nöjd
• Idag kommer han inte
• Sedan gick vi hem

If the subject came first, then it would be: så hon känner sig nöjd

Both patterns can be correct, but once nu is placed first, inversion happens.

Because känna sig is the normal way to say feel in Swedish when talking about your own state.

So:

• hon känner sig nöjd = she feels satisfied/pleased
• jag känner mig trött = I feel tired
• de känner sig glada = they feel happy

The word sig is the reflexive pronoun for third person, meaning something like herself here.

Without sig, känner often means know:

• Hon känner Anna = She knows Anna

So känner sig nöjd is a fixed and very common pattern.

Yes. Both are possible, but they are slightly different in feel.

• Hon är nöjd = She is satisfied / she is pleased
• Hon känner sig nöjd = She feels satisfied / she feels pleased

Känner sig emphasizes her personal feeling a bit more. In this sentence, that works well because the haircut has just led to that feeling.

Because adjectives agree with the noun or pronoun they describe.

Here the subject is hon, which is singular and common gender, so the adjective is nöjd.

Compare:

• hon är nöjd = she is satisfied
• det är nöjt = it is satisfied
• de är nöjda = they are satisfied

So nöjd is the correct form for hon.

Nöjd usually means satisfied, pleased, or content.

In this sentence, it probably means she is happy with the result of the haircut. It is often a little calmer and more specific than glad.

So:

• glad = happy, glad
• nöjd = satisfied, pleased, happy with something

After a haircut, nöjd is a very natural choice.

Har fått is the present perfect, and it connects a past action with the present situation.

That fits this sentence well:

• the haircut happened before now
• the result matters now
• she feels pleased now

So the present perfect is natural because the present result is important.

If you said Hon fick håret klippt, it would sound more like a past event being told as part of a story, with less focus on the present result.