Learn Hardcore Icelandic: Áhugamál hennar eru að lesa bækur og að dansa. - Her hobbies are reading books and dancing.

In this sentence it behaves as a plural (“hobbies”), which you see from the verb eru (“are”). For many neuter nouns ending in -mál, the singular and plural nominative forms look the same, so you need the verb to tell you if it’s singular or plural:

Áhugamál mitt er… = My hobby is… (singular)
Áhugamál mín eru… = My hobbies are… (plural)

Why is it hennar and not hún or henni?

Icelandic uses different forms of the pronoun “she” depending on the grammatical case:

hún = she (nominative, subject)
henni = to/for her (dative, indirect object)
hennar = her / of her (genitive, possession)

In Áhugamál hennar, we are expressing possession (“her hobbies”), so we need the genitive form hennar.

Also notice that Icelandic typically puts the possessive after the noun:

áhugamál hennar = her hobbies
You can say hennar áhugamál, but that has a slightly different emphasis (see below on word order).

Why is the verb eru and not er?

Icelandic, like English, changes the verb “to be” depending on singular vs. plural:

er = is (3rd person singular)
eru = are (3rd person plural)

Here, the subject is áhugamál hennar = her hobbies (plural idea), so the verb must agree and be eru:

Áhugamál hennar eru… = Her hobbies are…
If it were just one hobby:
Áhugamál hennar er… = Her hobby is…

What is the function of að in að lesa and að dansa?

Að in this context is the marker of the infinitive of the verb, like English “to” in “to read, to dance”:

að lesa = to read
að dansa = to dance

In English you usually say “Her hobbies are reading books and dancing” using -ing forms (gerunds). Icelandic prefers infinitives with að to express this kind of hobby or activity:

Áhugamál hennar eru að lesa bækur og að dansa.
Literally: Her hobbies are to read books and to dance.

Do I really need to repeat að before dansa, or can I say að lesa bækur og dansa?

Both forms are possible:

Áhugamál hennar eru að lesa bækur og að dansa.
Áhugamál hennar eru að lesa bækur og dansa.

Repeating að (version 1) sounds a bit more careful / explicit and is very clear.
Omitting the second að (version 2) is natural and common, especially in speech, just like English can say “to read and dance” with only one “to”.

So:

You can drop the second að.
Keeping it is also correct; the sentence you’re given is perfectly natural Icelandic.

Why is bækur plural? Could I say bók instead?

Bækur is the plural of bók (“book”):

bók = book (singular)
bækur = books (plural, nominative/accusative)

It’s an irregular plural with vowel change (ó → æ).

In this sentence, the idea is that she likes reading books in general, not just a single specific book, so the plural is more natural:

að lesa bækur = to read books

You could say að lesa bók = “to read a book”, but that would sound more like one book as an activity, which is not the usual way to describe a hobby.

What case is bækur in here, and why doesn’t it change form from nominative?

In this sentence, bækur is the direct object of að lesa (“to read”), so it is in the accusative plural.

For this noun, nominative plural and accusative plural happen to be identical:

nominative plural: bækur
accusative plural: bækur

So the form doesn’t change, but functionally here it is accusative (object of the verb lesa).

Why don’t we say Áhugamálin hennar with a definite ending, like “the” hobbies?

Icelandic often does not require the definite article where English uses “the” or treats something as specific.

Both are grammatically possible:

Áhugamál hennar eru…
Áhugamálin hennar eru…

Rough tendencies:

Áhugamál hennar = a bit more neutral / general: “Her hobbies are…”
Áhugamálin hennar = a bit more definite / specific, like “Her (particular set of) hobbies are…”, maybe already known from context.

In everyday use, Áhugamál hennar eru… is very natural and doesn’t feel “indefinite” or wrong to a native speaker. You can think of it as the normal way to introduce someone’s hobbies.

Can the word order be Hennar áhugamál eru að lesa bækur og að dansa instead?

Yes, that order is also grammatically correct:

Áhugamál hennar eru…
Hennar áhugamál eru…

The difference is one of emphasis:

Áhugamál hennar… = neutral, ordinary word order; focus mainly on the hobbies.
Hennar áhugamál… puts a bit more emphasis on “her” – for example, in contrast to someone else’s:
- Hennar áhugamál eru að lesa bækur og að dansa, en áhugamál bróður hennar eru að spila fótbolta.
  Her hobbies are reading books and dancing, but her brother’s hobbies are playing football.

Why is it lesa and dansa, not les or dansar?

Lesa and dansa are the infinitive forms:

lesa = to read
dansa = to dance

Les and dansar are finite present tense forms:

ég les = I read
hann dansar = he dances

After að used in this way, Icelandic requires the infinitive, just like English uses “to read, to dance”:

að lesa bækur = to read books
að dansa = to dance

Using les or dansar here would be grammatically wrong.

How would I say “His hobbies are reading books and dancing” or “My hobbies are reading books and dancing”?

You keep the structure and just change the possessive:

His hobbies are reading books and dancing.
Áhugamál hans eru að lesa bækur og að dansa.
- hans = his (genitive of hann, he)
My hobbies are reading books and dancing.
Áhugamál mín eru að lesa bækur og að dansa.
- mín = my (plural form agreeing with áhugamál)

How do you roughly pronounce Áhugamál hennar eru að lesa bækur og að dansa?

A broad, learner-friendly approximation (not strict IPA) would be:

Áhugamál ≈ OW-hoo-ga-maul
- Á like English ow in cow
- stress on the first syllable: ÁH-
hennar ≈ HEN-nar
eru ≈ EH-ru (the r is tapped/flapped)
að ≈ between ath (as in that, but with softer th) and a short a; often sounds like a short a.
lesa ≈ LE-sa (short e as in let)
bækur ≈ BIE-kur
- æ like eye
- kur with a short u (like put) and tapped r
og ≈ oh (often very short, almost like just o)
dansa ≈ DAHN-sa
- a as in father but short
- stress on DAN-

So the whole sentence, very roughly:

OW-hoo-ga-maul HEN-nar EH-ru a(ð) LE-sa BIE-kur oh a(ð) DAHN-sa.