Hún kyssti eiginmann sinn aftur þegar hann kom heim seint.

Because eiginmann sinn is the accusative singular form.

The verb kyssa means to kiss, and its object goes in the accusative case. So her husband has to appear in the accusative:

• nominative: eiginmaður
• accusative: eiginmann

The pronoun sinn also changes form to match the noun it belongs to. Since eiginmann is:

• masculine
• singular
• accusative

the matching form is sinn.

So:

• eiginmaður sinn = not correct here
• eiginmann sinn = correct, because it is the direct object of kyssti

Because sinn is the normal Icelandic reflexive possessive, used when the possessor is the subject of the same clause.

Here, the subject is Hún (she), and the husband belongs to her, the subject of the clause:

• Hún kyssti eiginmann sinn = She kissed her own husband

If you used hennar instead:

• Hún kyssti eiginmann hennar

that would usually mean:

• She kissed her husband = someone else’s husband, a different woman’s husband

So sinn shows that the husband belongs back to the subject hún.

In this sentence, aftur most naturally means again.

So the sentence means something like:

• She kissed her husband again when he came home late

In Icelandic, aftur can mean:

• again
• back

But in this context, with kyssti (kissed), again is the normal reading.

If the broader context were different, aftur could sometimes feel closer to in return or back, but that is not the most likely interpretation here.

Kyssti is the past tense of kyssa (to kiss).

So:

• að kyssa = to kiss
• hún kyssir = she kisses
• hún kyssti = she kissed

This tells you the action happened in the past.

The whole sentence is in the past:

• Hún kyssti ...
• þegar hann kom heim seint

So both the kissing and his coming home are presented as past events.

Because Icelandic is a verb-second language in main clauses, but not in the same way inside subordinate clauses.

The word þegar here means when and introduces a subordinate clause:

• þegar hann kom heim seint = when he came home late

Inside that clause, the order is normal:

• hann = subject
• kom = verb
• heim = direction/home
• seint = late

So this is completely normal Icelandic word order for a subordinate clause.

Because heim usually expresses motion toward home, while heima usually expresses location at home.

Compare:

• hann kom heim = he came home / he came to home
• hann var heima = he was at home

In your sentence, he is coming home, so the directional form heim is the right one.

Because seint is the adverb form, meaning late in the sense of arriving late.

Compare:

• seinn = adjective, used with a noun
• seint = adverb, used with a verb

Here it describes how/when he came home, not a noun:

• hann kom heim seint = he came home late

If you were describing a noun, you would use the adjective:

• seinn maður = a late man / a delayed man (depending on context)

But after kom in this sentence, the adverb seint is what you need.

Here, þegar means when.

So:

• þegar hann kom heim seint = when he came home late

In some contexts, þegar can feel quite close to when in a broad sense, and sometimes English translations may use once or something similar depending on context. But the basic meaning here is simply when.

It is not automatically as soon as. That stronger meaning would usually need context.

It most naturally describes the coming home.

So the sentence is understood as:

• She kissed her husband again when he came home late

not:

• She kissed her husband late

This is because seint is placed inside the þegar clause and follows kom heim, so it belongs naturally with that event:

• þegar hann kom heim seint

That means when he came home late.

Yes. Icelandic could also say:

• Þegar hann kom heim seint kyssti hún eiginmann sinn aftur.

That still means the same basic thing.

The main difference is emphasis and flow:

• starting with Hún kyssti ... puts focus on what she did
• starting with Þegar hann kom heim seint ... puts focus on the time/situation

In the version you were given, the sentence begins with the main action: Hún kyssti ...

Yes. Eiginmaður means husband, but literally it contains eigin, which means own.

Historically and structurally:

• eigin = own
• maður = man

So eiginmaður is the standard Icelandic word for husband.

Even though it literally contains the idea of own, Icelandic still uses the possessive pronoun:

• eiginmann sinn = her husband

So learners should treat eiginmaður simply as the normal word for husband, not as a phrase where eigin is being freely added each time.