Ég er þreyttur, þess vegna fer ég heim strax.

Yes. þreyttur is an adjective describing the subject ég (I), and Icelandic adjectives agree with the noun/pronoun they describe in gender, number, and case.

• ég refers to a masculine speaker → nominative singular masculine → þreyttur.
  If the speaker is feminine, you’d usually say: Ég er þreytt.
  (Neuter also uses þreytt in the nominative singular.)

Icelandic commonly uses vera (to be) + adjective to express states/conditions, including tiredness. So Ég er þreyttur is literally “I am tired” in structure.

þess vegna means therefore / that’s why / for that reason. It’s historically made of:

• þess = genitive of það (“that/of that”)
• vegna = “because of” (a word that governs the genitive)

Together they function as a fixed adverbial phrase meaning “for that reason.”

Because the sentence is effectively two clauses: 1) Ég er þreyttur
2) þess vegna fer ég heim strax
The comma marks the break before the connecting adverbial (þess vegna). In Icelandic, commas are often used to separate such clause-like units, especially when the second part is introduced by a connector like this.

This is the V2 rule (verb-second), which Icelandic follows in main clauses. If something other than the subject is placed first (here þess vegna), then the finite verb comes next, and the subject moves after it:

• First element: þess vegna
• Second element (finite verb): fer
• Then subject: ég

So you get þess vegna fer ég…, not þess vegna ég fer….

Yes, that’s common.

• þess vegna = “therefore / for that reason” (a bit more explicit, slightly more formal/clear)
• svo = “so” (often more conversational)
  Both typically trigger the same V2 word order: svo fer ég… / þess vegna fer ég…

fara is the infinitive (“to go”). fer is the present tense, 1st/3rd person singular form (“go/goes”):

• ég fer = “I go / I’m going”

heim can function as an adverb meaning home (wards), implying movement to one’s home. With verbs of motion, Icelandic often uses:

• fara heim = “go home”
  You can use other phrasing in specific contexts (e.g., fara til Íslands = “go to Iceland”), but heim is the natural word for “go home.”

strax (“immediately/right away”) is flexible, but typical placements are:

• End: … fer ég heim strax. (very common)
• After the verb: … fer ég strax heim. (also common; can emphasize “immediately”)
  Both are generally fine; word order can slightly change emphasis.

It usually implies right now / at the moment, because to be tired is a temporary state. Context can broaden it, but by default it’s understood as current.

In standard Icelandic, you normally keep the subject: þess vegna fer ég…
Icelandic is not generally a “drop-the-subject” language like Spanish/Italian. In casual speech you might sometimes omit things in fragments, but in full sentences you typically include ég.

You’d change the verbs:

• Ég var þreyttur, þess vegna fór ég heim strax.
  Here:
• er → var (was)
• fer → fór (went)

Adjective agreement stays the same (unless the speaker’s gender/number changes).