Ég kem strax heim eftir vinnu.

In Icelandic, the present tense is very commonly used for near-future plans when the context makes the timing clear.
Here, eftir vinnu (after work) already places the action in the future, so Ég kem… naturally means I’m coming / I’ll come….

Kem is the 1st person singular present tense of the verb koma (to come).
Basic present tense forms:

• ég kem = I come / I’m coming
• þú kemur = you come
• hann/hún/það kemur = he/she/it comes
• við komum = we come
• þið komið = you (pl.) come
• þeir/þær/þau koma = they come

Usually no. Icelandic normally keeps subject pronouns; the verb ending alone typically isn’t treated as enough information in everyday speech.
So Ég kem strax heim… is the standard way. (In very informal contexts you might drop it, but it’s not the safe default.)

The neutral pattern is:

• Subject + verb + adverbs + (direction/place)

So:

• Ég kem (subject + verb)
• strax (adverb: immediately)
• heim (direction: home)

Ég kem strax heim is the natural, unmarked order.

Strax means immediately / straight away / right away.
In this sentence it usually means: after work, there’s no delay—you go home immediately.

• heim = (to) home (motion/direction)
• heima = at home (location)

So:

• Ég kem heim = I come (back) home
• Ég er heima = I am at home

This sentence needs heim because it’s about coming to home.

Eftir often takes the accusative when it means after in time expressions.
Vinna (work) is feminine, and its accusative singular is vinnu, so:

• eftir vinnu = after work

(Conveniently, vinnu also looks the same as the dative singular, but here the intended use is the temporal after meaning.)

It can mean either depending on context. Without a possessive, Icelandic often leaves it general:

• eftir vinnu = after work / after (my) work

If you want to be explicit, you can say:

• eftir vinnuna = after the work (more specific, like a particular work shift/task)
• eftir vinnuna mína = after my work (very explicit)

Yes, but then Icelandic word order requires verb-second (V2): the verb comes immediately after the first element.

• Eftir vinnu kem ég strax heim.

Notice how kem stays second, and ég moves after the verb.

A rough guide:

• Ég ≈ yeh(g) (the g is often very soft or barely heard)
• kem ≈ kem
• strax ≈ strahks
• heim ≈ heym
• eftir ≈ eh-thir (the f sounds like f, and i is like i in bit)
• vinnu ≈ VIN-nu (short i, double n sound)