Hún snýr sér við þegar ég kalla á hana.

It is best learned as a set expression: snúa sér við = to turn around.

• snýr = turns
• sér = oneself / herself / himself (reflexive)
• við helps make the meaning around / back

So Hún snýr sér við means She turns around.

Sér is a reflexive pronoun. It shows that the action is directed back to the subject.

So:

• Hún snýr sér við = She turns herself around → natural English: She turns around

Icelandic often uses a reflexive pronoun where English simply uses turn around without herself.

Sér is the dative reflexive pronoun.

The verb expression snúa sér við requires this form. This is something learners usually need to memorize as part of the whole phrase.

A useful way to learn it is not as separate pieces, but as:

• ég sný mér við
• þú snýrð þér við
• hann/hún snýr sér við

So yes, sér is dative here because the idiom snúa sér við takes a reflexive pronoun in the dative.

Here við is not the preposition with. In this sentence, it is part of the verb phrase snúa sér við.

So you should think of:

• snúa sér við = turn around

rather than translating each word separately.

This is very common in Icelandic: a small word like við, upp, út, inn, etc. can combine with a verb to create a specific meaning.

Because hana is the accusative form of hún.

In the sentence:

• Hún = subject of the main clause: she
• hana = object of kalla á in the subordinate clause: her

So the structure is:

• Hún snýr sér við = She turns around
• þegar ég kalla á hana = when I call to her / call for her

English uses she/her in the same way, and Icelandic does too:

• hún = she
• hana = her

Because they belong to different parts of the sentence and have different grammatical roles.

• In the main clause, she is the subject: Hún snýr sér við
• In the þegar clause, she is the object: ég kalla á hana

So even though it is the same person, Icelandic changes the pronoun form based on case and function.

Kalla á means something like:

• call to
• call for
• call out to
• sometimes summon

In this sentence, þegar ég kalla á hana most naturally means when I call to her or when I call out to her.

It is different from just kalla, because á is an important part of the expression here.

They mean different things.

• kalla á hana = call to her / call for her
• kalla hana ... can mean call her ... / name her ...

For example:

• Ég kalla á hana. = I call to her.
• Ég kalla hana Önnu. = I call her Anna.

So the preposition á changes the meaning.

Because þegar introduces a subordinate clause: when I call to her.

In this kind of clause, Icelandic usually has straightforward subject-verb order:

• þegar ég kalla á hana

That is:

• þegar = when
• ég = I
• kalla = call
• á hana = to her / for her

This is more like English word order than the usual Icelandic main-clause verb-second pattern.

Snýr is the 3rd person singular present form of snúa.

The subject is hún = she, so the verb must match that:

• ég sný
• þú snýrð
• hún snýr

So Hún snýr sér við means She turns around.

Yes. Þegar very often means when, but in some contexts it can also relate to as soon as or timing more generally.

In this sentence, the natural meaning is simply:

• when I call to her

That is the meaning you should take here.

By itself, it can be either.

Hún snýr sér við þegar ég kalla á hana can mean:

• She turns around when I call to her
  which could describe a general habit

or, depending on context:

• She turns around when I call to her
  referring to a particular situation

The present tense in Icelandic, like in English, can often be used for either a general pattern or an event being described in context.

A few natural translations are:

• She turns around when I call to her.
• She turns around when I call her.
• She turns back when I call out to her.

The most literal grammar-friendly understanding is probably:

• She turns around when I call to her.

But in everyday English, when I call her may also be used, depending on context.