Mig grunar að hún sé þegar komin heim.

Because gruna is commonly used as an impersonal verb in Icelandic.

That means the person who has the suspicion is not put in the nominative as a normal subject. Instead, that person appears in the accusative:

• mig grunar = I suspect
• þig grunar = you suspect
• okkur grunar = we suspect

So mig is not a direct translation of English I; it is part of the Icelandic pattern used with this verb.

For the same reason: gruna is used impersonally here.

Since there is no normal nominative subject like ég, the verb appears in the default 3rd person singular form:

• mig grunar
• þig grunar
• hana grunar

The verb does not change to match mig.

Að here means that.

It introduces the subordinate clause:

• að hún sé þegar komin heim = that she is / has already come home

So the sentence structure is basically:

• Mig grunar = I suspect
• að ... = that ...

Sé is the present subjunctive of vera.

Icelandic often uses the subjunctive in subordinate clauses when the speaker is talking about something uncertain, suspected, possible, or not presented as a confirmed fact. Since Mig grunar means the speaker is only suspecting this, sé is very natural.

So:

• hún sé komin heim carries the idea she may be / I suspect she is home
• hún er komin heim sounds more like a straightforward fact

You may hear the indicative er in everyday speech, but sé is the more standard and grammatically expected choice when the content is presented as uncertain or only suspected.

So for a learner, sé is the safest form to use in this sentence.

Komin is the past participle of koma, used with vera.

The pattern vera kominn/komin/komið often means have arrived or be here now after arriving.

So:

• hún er komin heim can mean she has come home
• but it often strongly suggests she is home now

This is slightly different from a simple English-style perfect.

Because the participle agrees with the subject, and the subject here is hún, which is feminine singular.

So the forms are:

• hann er kominn
• hún er komin
• það er komið

And in the plural:

• þeir eru komnir
• þær eru komnar
• þau eru komin

In your sentence, hún requires komin.

These are close, but they are not exactly the same.

• sé komin heim focuses on the resulting state: she has arrived and is home now
• hafi komið heim is a more neutral has come home

So þegar komin heim strongly suggests that the speaker suspects she is already home now, not just that the action of coming home happened at some point.

Yes, here þegar means already.

That word can mean both already and when, but context tells you which one is meant.

In this sentence:

• hún sé þegar komin heim = she is already home / has already come home

It does not mean when here.

Because heim is used for movement toward home, while heima means at home.

Compare:

• fara heim = go home
• koma heim = come home
• vera heima = be at home

In komin heim, the idea is that she has arrived home, so heim is the correct form.

A useful way to divide it is:

• Mig grunar = I suspect
• að = that
• hún sé = she be
• þegar komin heim = already come/arrived home

A few things are happening:

1. In the main clause, mig comes first and the finite verb grunar follows.
2. Að starts the subordinate clause.
3. In that subordinate clause, hún is the subject and sé is the finite verb.
4. Þegar comes before komin and means already.
5. Heim comes last as the destination adverb.

So the sentence is built very naturally for Icelandic, even if it looks a bit different from English word order.