Dóttir mín er dugleg í skólanum.

In Icelandic, possessive words like minn / mín / mitt often come after the noun: dóttir mín = my daughter.

With family words, this is especially common and natural:

• móðir mín = my mother
• faðir minn = my father
• dóttir mín = my daughter

Putting the possessive before the noun is possible in some contexts, but the post-noun position is the normal pattern here.

Because the possessive has to agree with the noun it belongs to.

Dóttir is:

• feminine
• singular
• nominative here

So the correct form is mín.

Compare:

• minn = masculine
• mín = feminine
• mitt = neuter

So:

• sonur minn = my son
• dóttir mín = my daughter
• barn mitt = my child

It is in the nominative case, because it is the subject of the sentence.

In Dóttir mín er dugleg í skólanum, the subject is Dóttir mín. Since the sentence is saying something about her, nominative is the expected case.

That is also why the words agreeing with it are nominative too:

• mín
• dugleg

Because adjectives also have to agree with the noun they describe.

Here, dugleg describes dóttir, which is:

• feminine
• singular
• nominative

So the adjective must also be feminine singular nominative: dugleg.

Compare:

• Sonur minn er duglegur. = My son is hardworking.
• Dóttir mín er dugleg. = My daughter is hardworking.
• Barnið mitt er duglegt. = My child is hardworking.

Icelandic usually puts the definite article onto the end of the noun instead of using a separate word like English the.

So:

• skóli = school
• skólanum = in/at the school (dative singular definite)

In this sentence, the is built into skólanum.

Because í can take different cases depending on meaning.

• í
  • dative = location, in / at
• í
  • accusative = motion, into

Here the meaning is location: she is hardworking in/at school, not moving into school. So Icelandic uses the dative:

• í skólanum = in/at the school

Compare:

• Hún er í skólanum. = She is in/at the school.
• Hún fer í skólann. = She goes into/to the school.

Yes. Very often í skólanum is understood naturally as at school or in school, depending on context.

So even though í literally means in, the phrase can be used in the broader everyday sense of someone being at school or in the school environment.

Because dugleg is being used as a predicate adjective. It comes after the verb er and describes the subject.

So:

• Dóttir mín er dugleg. = My daughter is hardworking.

If you put the adjective directly before the noun, that is a different structure:

• dugleg dóttir = a hardworking daughter

Both are correct, but they work differently:

• er dugleg = is hardworking
• dugleg dóttir = hardworking daughter

Yes, at least in usage.

With close family terms, Icelandic very often uses the noun without the definite article:

• dóttir mín
• móðir mín
• bróðir minn

That is the natural pattern in a sentence like this.

A form with the article can sometimes appear in other contexts, but dóttir mín is the standard and most straightforward way to say my daughter here.

It often means something like:

• hardworking
• diligent
• good about doing what one should
• capable / industrious

In school contexts, dugleg í skólanum usually suggests that someone is doing well, working hard, or being a diligent student.

So it is broader than just one English word, but hardworking or diligent is usually a good match here.