Hann er farinn heim.

Because Hann er farinn heim does not work exactly like a plain simple past such as Hann fór heim.

• Hann fór heim = He went home.
  This simply reports the action.

• Hann er farinn heim = He has gone home / He is gone home.
  This stresses the resulting state: he has left, and he is now away.

So er farinn is often used when the important idea is that the person is already gone.

Farinn is the past participle of fara (to go).

In this sentence, it is used with er (from vera, to be) to form a meaning like has gone or is gone.

It also behaves a lot like an adjective, which is why it changes form depending on who is being talked about.

For example:

• Hann er farinn heim. — masculine
• Hún er farin heim. — feminine
• Þau eru farin heim. — neuter plural/mixed group in some contexts
• Þeir eru farnir heim. — masculine plural

So farinn is not a fixed form; it agrees with the subject.

Because it agrees with hann, which is:

• masculine
• singular
• nominative

So the participle appears in the matching form farinn.

If the subject changed, the participle would change too:

• Hún er farin heim. — she
• Barnið er farið heim. — the child / it
• Þeir eru farnir heim. — they (masculine or mixed group)
• Þær eru farnar heim. — they (all feminine)

This agreement is a very common feature of Icelandic.

Because heim is a special directional word meaning homeward / home.

With verbs of motion, Icelandic often says simply:

• fara heim — go home
• koma heim — come home
• keyra heim — drive home

So heim already contains the idea of movement toward home, and no extra preposition is needed.

This is similar to English go home, where we also usually do not say go to home.

This is an important distinction:

• heim = to home / homeward → direction, movement
• heima = at home → location, no movement

Examples:

• Hann fer heim. — He goes home.
• Hann er heima. — He is at home.

So in Hann er farinn heim, heim is used because the sentence refers to going somewhere, not just being located there.

Not quite.

Both can often be translated naturally into English, but the nuance is different:

• Hann fór heim focuses on the event: he went home.
• Hann er farinn heim focuses on the present result: he has gone home / he is already gone home.

If someone asks where he is now, Hann er farinn heim is especially natural because it suggests he is no longer here.

Yes.

Hann er farinn by itself can mean something like:

• He has left
• He is gone

In that case, the destination is not stated.

Adding heim tells you where he went:

• Hann er farinn. — He has left.
• Hann er farinn heim. — He has gone home.

This is because Icelandic forms some gone/come/arrived-type result expressions with vera (to be) plus a participle.

So with motion verbs, Icelandic often uses a structure that feels more like be gone than English have gone.

That is why:

• er farinn literally looks like is gone
• but in English it is often best translated as has gone

This is one of those places where Icelandic grammar does not match English word-for-word.

Yes.

A yes/no question is:

• Er hann farinn heim? — Has he gone home? / Is he gone home?

Icelandic commonly forms yes/no questions by putting the finite verb first.

Compare:

• Hann er farinn heim. — statement
• Er hann farinn heim? — question

The verb er / eru and the participle farinn must match the subject.

Examples:

• Ég er farinn heim. — I have gone home (said by a man)
• Ég er farin heim. — same, said by a woman
• Þú ert farinn heim. — You have gone home (to a man)
• Þú ert farin heim. — You have gone home (to a woman)
• Hún er farin heim. — She has gone home
• Þeir eru farnir heim. — They have gone home
• Þær eru farnar heim. — They have gone home (all female)

This kind of gender and number agreement is something English speakers often need time to get used to.

Yes, very common.

Native speakers often use this type of sentence when talking about someone who has already left or is no longer present.

For example, if someone asks where Jón is, a natural answer could be:

• Hann er farinn heim.

It sounds very normal and idiomatic in Icelandic, especially when the current result matters more than the past event itself.