Dóttir hennar situr í aftursætinu og syngur á leiðinni í skólann.

Both are possible, but they do not sound exactly the same.

• dóttir hennar = her daughter in a neutral, very common Icelandic pattern
• hennar dóttir can sound more contrastive or emphatic, something like her daughter (as opposed to someone else’s)

Icelandic often places possessive pronouns like hennar, hans, þeirra after the noun.

So in ordinary speech, dóttir hennar is a very natural way to say her daughter.

Because Icelandic usually does not combine the definite article and a possessive in the same way English does.

• dóttir hennar already means her daughter
• dóttirin hennar means something more like the daughter of hers or her daughter with extra definiteness/emphasis

In many everyday contexts, Icelandic prefers the simpler form:

• dóttir hennar = her daughter

So the sentence is using the most natural, basic structure.

It is in the nominative singular because it is the subject of the sentence.

The verb is situr = sits, and the person doing the action is dóttir hennar.

So:

• dóttir = nominative singular of dóttir (daughter)

Situr is the 3rd person singular present tense of the verb sitja (to sit).

Since the subject is daughter = one person, Icelandic uses singular:

• ég sit = I sit
• þú situr = you sit
• hann/hún/það situr = he/she/it sits

So:

• Dóttir hennar situr = Her daughter sits / is sitting

Because sitja í takes the dative when it describes location, not motion.

Here the meaning is sitting in the back seat, so it is a static location.

• aftursæti = back seat
• dative singular definite = aftursætinu

So:

• í aftursætinu = in the back seat

A useful rule:

• í + dative often = location, being in something
• í + accusative often = motion into something

It breaks down like this:

• aftursæti = back seat
  • aftur- = back / rear
  • sæti = seat
• -nu here is part of the ending for the definite dative singular of this neuter noun

So aftursætinu means:

• the back seat
  or in context:
• in the back seat

Syngur is the 3rd person singular present tense of syngja (to sing).

Because the same subject, dóttir hennar, is still being talked about, Icelandic uses another 3rd person singular verb:

• situr = sits
• syngur = sings

So:

• Dóttir hennar situr ... og syngur ...
• Her daughter sits ... and sings ...

Because Icelandic does not need to repeat the subject when the same subject continues across coordinated verbs.

So:

• Dóttir hennar situr í aftursætinu og syngur ...

means:

• Her daughter sits in the back seat and sings ...

English also often does this:

• Her daughter sits in the back seat and sings not
• Her daughter sits in the back seat and she sings

Á leiðinni is a very common expression meaning:

• on the way
• along the way

It comes from:

• leið = way, route
• á = on
• leiðinni = the dative singular definite form

So literally it is something like on the way, and idiomatically that is exactly how it is used.

In this sentence:

• á leiðinni í skólann = on the way to school

Because the two prepositional phrases are doing different jobs.

• á leiðinni is part of the idiomatic expression on the way
• í skólann expresses direction: to school

With í, Icelandic often uses:

• dative for location: in
• accusative for motion into/toward

Here there is movement toward the school, so Icelandic uses the accusative:

• skólann = accusative singular definite of skóli

So:

• á leiðinni í skólann = on the way to school

Because í here shows motion toward a destination, so it takes the accusative.

Compare:

• í skólanum = in the school / at the school (location, dative)
• í skólann = into the school / to school (motion, accusative)

In this sentence, the daughter is singing while going toward school, so Icelandic uses:

• í skólann

Yes, it is very normal in Icelandic.

Even though English says to school, Icelandic often uses í with places when there is movement toward them:

• fara í skólann = go to school
• fara í búðina = go to the shop
• fara í vinnuna = go to work

So í skólann is exactly what you would expect.

The sentence follows a very straightforward Icelandic main-clause pattern:

• Dóttir hennar = subject
• situr = verb
• í aftursætinu = place
• og syngur = coordinated second verb
• á leiðinni í skólann = time/circumstantial phrase

So the structure is basically:

• Subject + Verb + Place + and + Verb + Circumstantial phrase

This is a very natural word order for a simple declarative sentence.

Yes. Icelandic present tense often covers both:

• simple present: sits / sings
• present progressive sense: is sitting / is singing

So depending on context, the sentence could be understood as:

• Her daughter sits in the back seat and sings on the way to school or
• Her daughter is sitting in the back seat and singing on the way to school

English forces a clearer choice; Icelandic often does not.

In normal interpretation, it describes the overall situation: she is in the back seat and singing while they are on the way to school.

Semantically, it fits the whole scene, though it is placed after syngur. Icelandic often puts this kind of phrase later in the sentence even when it applies broadly.

So the natural reading is:

• Her daughter sits in the back seat and sings while on the way to school

not just that the singing happens on the way, while the sitting happens somewhere else.

Yes, very closely.

• og = and

It links the two verbs/actions:

• situr = sits
• syngur = sings

So:

• situr ... og syngur ... = sits ... and sings ...

This is one of the easiest parts of the sentence for an English speaker.