Dóttir mín klifrar upp á klifurgrindina í garðinum.

Both are possible, but dóttir mín is the more neutral and natural way to say my daughter in ordinary Icelandic.

Using mín dóttir usually sounds more emphatic, contrastive, literary, or emotionally marked, as if you were stressing my daughter rather than someone else’s.

So in a simple statement, dóttir mín is what learners should expect most often.

With many close family words, Icelandic often leaves out the definite article when a possessive follows.

So you commonly get:

• dóttir mín = my daughter
• móðir mín = my mother
• bróðir minn = my brother

With many other nouns, Icelandic more often uses the definite form before a possessive:

• bíllinn minn = my car
• bókin mín = my book

So dóttir mín is the normal idiomatic expression here.

They are nominative singular feminine.

Why:

• dóttir is the subject of the sentence
• the subject is in the nominative
• mín agrees with dóttir in gender, number, and case

So:

• dóttir = nominative singular feminine
• mín = nominative singular feminine to match it

Klifrar is the 3rd person singular present tense of klifra.

So:

• að klifra = to climb
• hún klifrar = she climbs / she is climbing

In Icelandic, the simple present can often translate either as:

• climbs
• is climbing

The exact English wording depends on context.

Upp á expresses movement up onto something.

That fits the idea of climbing onto a structure:

• klifra upp á klifurgrindina = climb up onto the climbing frame

Using just á would not express the upward movement as clearly. With verbs like klifra, upp á is very natural when someone is moving upward onto an object.

Because this is the definite accusative singular form of klifurgrind.

Breakdown:

• klifurgrind = climbing frame
• klifurgrindin = the climbing frame
• klifurgrindina = the climbing frame, as an accusative object/destination

Here it is accusative because upp á is showing motion onto something. Icelandic often uses:

• accusative for motion toward/onto/into
• dative for location on/in something

So:

• upp á klifurgrindina = onto the climbing frame
• á klifurgrindinni = on the climbing frame

Yes. Icelandic very often makes compounds as a single word.

Here:

• klifur = climbing
• grind = frame, rack, structure

Together:

• klifurgrind = climbing frame

Then Icelandic adds the article and case ending to the whole compound:

• klifurgrindina

This is extremely normal in Icelandic.

Because í can take either accusative or dative, depending on meaning.

A very important Icelandic pattern is:

• accusative after í = motion into
• dative after í = location in

Here, the daughter is climbing onto the frame, and that frame is in the garden. The garden is just the location of the scene, not the destination of movement.

So:

• í garðinum = in the garden, location
• í garðinn = into the garden, movement into it

It can naturally be understood as the location of the whole situation.

In practice, it means something like:

• the action happens in the garden
• the climbing frame is in the garden

This kind of small ambiguity is normal and works much like English. Context usually makes it clear enough.

No. Icelandic does not have an indefinite article like English a/an.

So a bare noun can mean:

• a daughter
• daughter depending on context

For definiteness, Icelandic usually adds the equivalent of the as a suffix:

• garður = garden
• garðurinn = the garden
• garðinum = in the garden

Likewise:

• klifurgrind = climbing frame
• klifurgrindina = the climbing frame

No. Icelandic usually follows a verb-second pattern.

So if you start with Í garðinum, the finite verb normally comes next:

• Í garðinum klifrar dóttir mín upp á klifurgrindina.

That is a very important feature of Icelandic word order. The verb tends to stay in the second position, even when something other than the subject comes first.