Min søster tager både sin guitar og sin sangbog med, når hun skal øve.

Because Danish uses a reflexive possessive when the owner is the subject of the same clause.

Here, the subject is Min søster, so sin guitar and sin sangbog mean her own guitar and her own songbook.

Why not the others?

• sin is used with a common-gender singular noun
• sit is used with a neuter singular noun
• sine is used with plural nouns
• hendes is usually used when the thing belongs to some other female person, not the subject herself

So:

• sin guitar because guitar is en guitar
• sin sangbog because sangbog is en sangbog

Compare:

• Hun tager sin guitar med = She takes her own guitar along
• Hun tager hendes guitar med = She takes another woman’s guitar along

Because in Danish, it is very normal to repeat the possessive before each singular noun in a list like this.

So sin guitar og sin sangbog sounds natural and clear.

If you left out the second sin, the sentence would sound less standard here. Danish usually wants each singular count noun to have its own determiner or possessive in this kind of coordination.

So this is the natural pattern:

• sin guitar og sin sangbog
• min jakke og min taske
• hans telefon og hans nøgler

Både ... og ... means both ... and ....

It connects two things and says that both are included.

Pattern:

• både X og Y

In your sentence:

• både sin guitar og sin sangbog = both her guitar and her songbook

Other examples:

• Hun taler både dansk og engelsk = She speaks both Danish and English
• Jeg købte både kaffe og te = I bought both coffee and tea

Because tage med is a verb-plus-particle expression, similar to English take along or bring along.

In Danish main clauses, the finite verb often appears early, while the particle can come later:

• Hun tager sin guitar med

So the pieces belong together in meaning:

• tage med = take along / bring along

This is very common in Danish with particles such as med, op, ud, ind, af, and so on.

Compare:

• Han tager bogen med
• Vi skriver det ned
• Hun går ud

Because når introduces a subordinate clause, and subordinate clauses normally have subject + finite verb order in Danish.

So:

• når hun skal øve is correct

Not:

• når skal hun øve

The verb-before-subject order happens in many main clauses, especially when something else comes first:

• I dag skal hun øve
• Når hun skal øve, tager min søster sin guitar med

In that full sentence, Når hun skal øve is the subordinate clause, and tager min søster ... is the main clause with inversion after the fronted clause.

Når here means when or whenever, and it is used for something general, repeated, or expected.

So når hun skal øve suggests something like:

• when she is going to practice
• whenever she has to practice

Why not the others?

• da is usually for a specific event in the past
• hvis means if

So:

• Når hun skal øve, tager hun sin guitar med = When/whenever she has to practice, she takes her guitar along
• Da hun skulle øve, tog hun sin guitar med = When she had to practice, she took her guitar along
• Hvis hun skal øve, tager hun sin guitar med = If she has to practice, she takes her guitar along

Not necessarily a strong must.

Skal can mean several things depending on context, including:

• must
• have to
• be supposed to
• be going to

In når hun skal øve, it often feels more natural in English as:

• when she has to practice
• when she is going to practice

So skal does not always sound as strong as English must.

Because øve can be used on its own to mean practice or rehearse, especially in musical contexts.

So:

• hun skal øve = she has to practice / rehearse

You can also see:

• øve noget = practice something
  • Hun øver en ny sang
• øve sig = practice, often with emphasis on improving oneself at something
  • Hun øver sig i at synge

In this sentence, bare øve sounds natural because the context already makes it clear what kind of practice is meant.

Because Danish forms compound nouns very freely, and they are usually written as one word.

So:

• sang
  • bog = sangbog

The last part is the main noun, so a sangbog is a kind of bog.

This is extremely common in Danish:

• håndbog = handbook
• skolebog = schoolbook
• musiklærer = music teacher
• sommerferie = summer holiday

English often writes similar ideas as two words, but Danish usually joins them.

Because the possessive already does the job that an article would normally do.

So Danish says:

• min søster = my sister
• sin guitar = her own guitar

Not:

• den min søster
• den sin guitar

This works like English:

• my sister, not the my sister
• her guitar, not the her guitar

Yes. That is also correct.

Danish is a verb-second language in main clauses, so when the subordinate clause comes first, the main clause shows inversion:

• Når hun skal øve, tager min søster både sin guitar og sin sangbog med

If the main clause comes first, you get:

• Min søster tager både sin guitar og sin sangbog med, når hun skal øve

Both are grammatical. The difference is mainly what you want to emphasize first:

• first version: the when-clause
• second version: my sister and what she brings