Hún kastar boltanum til vinkonu sinnar, og stelpan grípur hann auðveldlega.

Because the verb kasta normally takes the thing being thrown in the dative case.

• bolti = ball
• boltinn = the ball (nominative)
• boltann = the ball (accusative)
• boltanum = the ball (dative)

So in Hún kastar boltanum..., boltanum is dative because of the verb kastar.

This is one of those things Icelandic learners often just have to learn with the verb:

• kasta einhverju = to throw something
  with eitthvað in the dative

Hann means it / him, and here it refers back to boltann / boltanum (the ball).

The important thing is that pronouns also change by case:

• boltanum is dative because of kasta
• hann is accusative because of grípa

The verb grípa takes a direct object in the accusative:

• grípa hann = catch it

Also, bolti is a masculine noun, so the pronoun is masculine too:

• hann = masculine accusative singular

So the case changes because the verb changes.

Because the preposition til requires the genitive case.

So:

• til = to / toward / to the place of, depending on context
• after til, the noun must be in the genitive

Here:

• vinkona = female friend
• after til, it becomes vinkonu

That may look confusing, because vinkonu is the same form used for more than one case. With this kind of feminine noun, the singular forms often look like this:

• nominative: vinkona
• accusative: vinkonu
• dative: vinkonu
• genitive: vinkonu

So in this sentence, vinkonu is genitive because til demands it, even though the form looks the same as accusative and dative.

Because sinn is the reflexive possessive in Icelandic. It refers back to the subject of the clause.

In this sentence, the subject is Hún (she), so:

• vinkonu sinnar = her own friend

If you said vinkonu hennar, that would usually mean:

• the friend of some other woman/girl, not the subject herself

So the contrast is:

• Hún kastar boltanum til vinkonu sinnar
  = She throws the ball to her own friend

• Hún kastar boltanum til vinkonu hennar
  = She throws the ball to her friend where her means some other female person

This is a very important Icelandic distinction.

Because sinnar has to agree with vinkonu in gender, number, and case.

Here, vinkonu is:

• feminine
• singular
• genitive

So sinn must also be in the feminine singular genitive form:

• sinnar

You can think of sinn as behaving a bit like an adjective or determiner that changes form to match the noun it belongs to.

They are different nouns:

• vinkona = female friend
• stelpa = girl

So:

• til vinkonu sinnar = to her friend
• stelpan = the girl

In context, stelpan is probably the same person as the friend, but the sentence uses a different noun to refer to her. That is possible in Icelandic, just as in English:

• She throws the ball to her friend, and the girl catches it easily.

So the sentence likely means the friend is the girl who catches the ball.

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

So:

• stelpa = girl
• stelpan = the girl

This is very common in Icelandic:

• bolti = ball
• boltinn = the ball
• stelpa = girl
• stelpan = the girl

So English often uses a separate article, but Icelandic usually uses a suffix.

They are both present tense, third person singular forms.

That matches the subjects:

• Hún = she
• stelpan = the girl

So:

• kasta = to throw
• kastar = throws

and:

• grípa = to catch
• grípur = catches

So the sentence is describing what is happening in the present:

• She throws...
• the girl catches...

Because auðveldlega is an adverb meaning easily, and this is a natural place for it in Icelandic.

It modifies the verb grípur:

• grípur hann auðveldlega = catches it easily

Placing the adverb at the end is a normal, neutral word order here. Icelandic word order can be flexible, but this version sounds natural and straightforward.

Yes, it could, but using stelpan makes the sentence clearer.

If you wrote:

• Hún kastar boltanum til vinkonu sinnar, og hún grípur hann auðveldlega

there would be two different feminine singular people in the sentence:

• the one who throws
• the friend

Then hún in the second clause could feel ambiguous.

Using stelpan helps the reader understand that the catcher is the girl being referred to, probably the friend.

Because Icelandic grammar relies heavily on case to show each word’s role in the sentence.

In this sentence:

• Hún = subject, nominative
• boltanum = object of kasta, dative
• til vinkonu sinnar = after til, genitive
• stelpan = subject, nominative
• hann = object of grípa, accusative

So even though English mostly uses word order to show meaning, Icelandic often uses endings as well. That is why learners see so many different forms in a single sentence.