Min søster er grundig, og desuden er hun meget pålidelig.

Because søster is a common-gender noun in Danish.

Danish possessives agree with the gender/number of the noun:

• min = for a singular common-gender noun
• mit = for a singular neuter noun
• mine = for plural nouns

So:

• min søster
• mit hus
• mine søstre

In Danish, a possessive like min usually replaces the article. So you say:

• min søster = my sister
• not min en søster or anything like that

This works the same way as in English: you say my sister, not my the sister.

Here the adjectives come after er, so they are predicative adjectives. In Danish, predicative adjectives still agree with the subject, but the form depends on gender/number.

Because søster / hun is singular common gender, you use the basic form:

• hun er grundig
• hun er pålidelig

Compare:

• huset er grundigt → neuter singular
• de er grundige → plural

So in your sentence, the base forms are correct.

Grundig usually means thorough, careful, or methodical. It often suggests that someone does things properly and pays attention to details.

So it is not just careful in the sense of avoiding danger. It more often means someone works in a complete, solid, well-organized way.

This is because of the Danish V2 rule: in a main clause, the finite verb normally comes in the second position.

In the second clause, desuden is placed first. That means the verb must come next:

• desuden er hun meget pålidelig

Not:

• desuden hun er meget pålidelig

This kind of inversion is very common in Danish.

Og simply links the two clauses: and.
Desuden adds an extra meaning like besides, furthermore, or in addition.

So together they mean something like:

• and, furthermore, ...
• and besides, ...

It makes the second quality sound like an additional important point, not just another item in a list.

Yes, that is also possible.

Both are grammatical, but the focus changes slightly:

• og desuden er hun meget pålidelig → stronger emphasis on desuden
• og hun er desuden meget pålidelig → slightly more neutral flow

In the original sentence, desuden is fronted, so it stands out more.

Because this sentence contains two coordinated main clauses:

1. Min søster er grundig
2. og desuden er hun meget pålidelig

Each main clause needs its own finite verb. Danish normally does not leave out er here.

So the repetition is completely natural.

Meget is an adverb here, meaning very. It modifies the adjective pålidelig:

• meget pålidelig = very reliable

It does not change form here. Danish commonly uses meget before adjectives in exactly this way.

Yes, Danish could repeat min søster, but using hun is more natural and less repetitive.

So:

• Min søster er grundig, og desuden er hun meget pålidelig.

sounds smoother than repeating:

• Min søster er grundig, og desuden er min søster meget pålidelig.

Just like in English, once the person is clear, a pronoun is usually preferred.

A rough guide:

• søster sounds approximately like SUR-sta
  
  • the ø sound does not exist exactly in English, but it is somewhat like the vowel in French deux or German schön
• pålidelig is roughly POH-lee-de-lai or POH-lee-thə-lai, depending on speaker and accent

A few helpful notes:

• Danish pronunciation is often less phonetic than spelling suggests.
• The d in words like pålidelig is often softened.
• Final syllables are often reduced in everyday speech.

In modern Danish, the comma before og in a sentence like this is often optional, depending on the comma system and style being used.

So you may see both:

• Min søster er grundig, og desuden er hun meget pålidelig.
• Min søster er grundig og desuden er hun meget pålidelig.

The version with the comma is clear and perfectly acceptable.