Hun vil gerne få farvet håret hos den samme frisør, hvis der er en ledig tid.

Hun vil gerne means something like she would like to or she wants to.

• vil = wants to / will
• gerne adds the sense of gladly / willingly / would like to

So:

• Hun vil få farvet håret = She wants to get her hair dyed
• Hun vil gerne få farvet håret = She would like to get her hair dyed

In Danish, gerne is very commonly used to make a wish or desire sound more natural and polite.

This is a very common Danish pattern:

• få + past participle + object

It often means to get something done.

So:

• få farvet håret = get one’s hair dyed
• literally: get dyed the hair

Other examples:

• få klippet håret = get one’s hair cut
• få repareret bilen = get the car repaired
• få taget et billede = get a photo taken

This structure is especially useful when someone else performs the action.

Because farvet is the past participle of farve (to dye / to color).

In the pattern få + past participle, Danish uses the participle:

• farve = to dye
• farvet = dyed

So:

• få farvet håret = get the hair dyed

This is similar to English:

• to dye → to get the hair dyed

Danish often uses the definite noun where English would use a possessive.

So instead of saying:

• her hair

Danish often says:

• the hair = håret

when it is already obvious whose hair it is.

Because the subject is hun (she), it is naturally understood that it is her hair.

So:

• Hun vil gerne få farvet håret
  literally: She would like to get the hair dyed naturally: She would like to get her hair dyed

This is very common with body parts and personal items:

• Han vasker hænderne = He washes his hands
• Hun børster tænderne = She brushes her teeth

Hos often means at, with, or in the care of, especially with people and professionals.

So:

• hos frisøren = at the hairdresser’s / with the hairdresser
• hos lægen = at the doctor’s
• hos tandlægen = at the dentist’s

In this sentence:

• hos den samme frisør = with the same hairdresser

It emphasizes that she wants to go to the same professional as before.

In Danish, samme usually appears with den/det/de when it modifies a definite noun phrase.

So:

• den samme frisør = the same hairdresser
• det samme sted = the same place
• de samme mennesker = the same people

Using just samme frisør would usually not sound right here.

This means:

• if there is an available appointment / if there is a free time slot

Breakdown:

• hvis = if
• der er = there is
• en ledig tid = a free/available time

Der er is the normal Danish way to say there is / there are.

Examples:

• Der er en stol her. = There is a chair here.
• Der er mange mennesker. = There are many people.

So here:

• hvis der er en ledig tid = if there is an available appointment

It is understandable and works, but in everyday Danish, people often say things like:

• en ledig tid
• en ledig time
• en tid
• en ledig aftale
• en ledig booking (more modern/business-like in some contexts)

At a salon or doctor’s office, en ledig tid is very common and natural. It means a free opening in the schedule.

So yes, it is a good and natural phrase here.

Good question. Danish does change word order in subordinate clauses, but what changes most clearly is the position of things like ikke and some adverbs.

Main clause:

• Hun kommer ikke.

Subordinate clause:

• ... hvis hun ikke kommer.

In your sentence:

• hvis der er en ledig tid

there is no adverb like ikke, so the word order may look similar to English. But this is still a subordinate clause introduced by hvis.

A good comparison:

• Main clause: Der er en ledig tid.
• Subordinate clause: ... hvis der er en ledig tid.

Because there is no adverb to reveal the change, the structure looks simple here.

Yes, but the meaning changes.

• Hun vil gerne få farvet håret ...
  = She would like to get her hair dyed ...
• Hun kan få farvet håret ...
  = She can get her hair dyed ... / It is possible for her to get her hair dyed ...

So:

• vil gerne expresses desire
• kan expresses possibility/ability

If the point is what she wants, vil gerne is the right choice.

Yes, have farvet håret is also possible in Danish, and in many situations it means almost the same thing.

• få farvet håret = get her hair dyed
• have farvet håret = literally have her hair dyed

In practice, both can be used for salon situations. However:

• få strongly highlights receiving / getting something done
• have can sound a bit more like arranging to have something done

For many learners, få + past participle is the safest and most common pattern to learn first.

It is grammatically built with vil, which can express either:

• want/desire
• or sometimes future

In this sentence, vil gerne is mainly about desire, not a plain future.

So the sense is:

• She would like to get her hair dyed with the same hairdresser, if there is an available appointment.

It is not simply a neutral future statement like She will get her hair dyed.

The main clause is:

• Hun vil gerne få farvet håret hos den samme frisør

The subordinate clause is:

• hvis der er en ledig tid

The subordinate clause is introduced by hvis (if) and gives the condition for the main clause.

So the sentence structure is:

• Main idea: She would like to get her hair dyed with the same hairdresser
• Condition: if there is an available appointment

Here are the main ones:

• hun = she
• ville / vil = want to, will
• gerne = gladly, would like to
• få = get, receive
• farve = to dye, color
• hår = hair
• hos = at, with
• samme = same
• frisør = hairdresser
• hvis = if
• der er = there is
• ledig = free, available
• tid = time; in this context, an appointment slot

This is a useful sentence because it contains several very common Danish patterns learners will see again and again.