Ég heng skyrtuna á herðatréð og set hana í skápinn.

Because hengja and setja are the infinitive forms, meaning to hang and to put/set.

In the sentence, the verbs are conjugated for 1st person singular present:

• ég heng = I hang
• ég set = I put

So the full pattern here is:

• að hengja → ég heng
• að setja → ég set

Skyrtuna is accusative singular definite.

It is in the accusative because it is the direct object of the verb heng. In other words, it is the thing being hung.

The basic noun is:

• skyrta = shirt

Its forms here are roughly:

• skyrta = a shirt
• skyrtuna = the shirt, as a direct object

Because skyrta is a feminine noun.

In Icelandic, pronouns like he / she / it often follow the grammatical gender of the noun, not whether the thing is biologically male or female.

So since skyrta is feminine, the pronoun referring back to it is feminine too:

• hún = she / it
• hana = her / it (accusative)

Here, hana is accusative because it is the direct object of set.

Because both phrases describe movement toward a place:

• á herðatréð = onto the hanger
• í skápinn = into the closet

With prepositions like á and í, Icelandic often uses:

• accusative for motion/direction
• dative for location/stillness

So in this sentence, the shirt is being moved:

• onto the hanger
• into the closet

That is why you get herðatréð and skápinn here.

Then Icelandic would normally use the dative after á and í.

Compare:

• á herðatréð = onto the hanger
• á herðatrénu = on the hanger

and

• í skápinn = into the closet
• í skápnum = in the closet

So the case helps show the difference between:

• direction/change
• position/location

This is one of the most important things to notice in this sentence.

Because Icelandic usually expresses the as an ending attached to the noun.

So instead of a separate word, the noun itself changes:

• skyrta = a shirt
• skyrtan / skyrtuna = the shirt
• skápur = a closet
• skápinn = the closet
• herðatré = a hanger
• herðatréð = the hanger

This attached definite article changes depending on gender, number, and case.

Because the ending for the is not one fixed form in Icelandic. It changes according to the noun’s:

• gender
• number
• case

In this sentence:

• skyrtuna is feminine accusative singular definite
• herðatréð is neuter singular definite
• skápinn is masculine accusative singular definite

So the different endings are normal. They reflect the grammar of each noun, not a difference in meaning.

They are:

• skyrta — feminine
• herðatré — neuter
• skápur — masculine

That helps explain the forms used in the sentence:

• skyrtuna comes from skyrta
• herðatréð comes from herðatré
• skápinn comes from skápur

Learning nouns together with their gender is very important in Icelandic.

Because many neuter nouns have the same form in nominative and accusative, especially in the singular.

So even though herðatréð is governed by á with movement and is therefore accusative, its form looks the same as the nominative definite form.

That can feel strange to an English speaker, but it is very common in Icelandic.

This is a very useful distinction:

• hengja = to hang something
  
  • transitive
• hanga = to be hanging
  
  • intransitive

So:

• Ég heng skyrtuna á herðatréð = I hang the shirt on the hanger
• Skyrtan hangir á herðatrénu = The shirt is hanging on the hanger

English often uses hang for both ideas, but Icelandic separates them more clearly.

Normally, no — not in standard learner-style Icelandic.

The verb setja needs an object here, and hana tells you what is being put into the closet. It refers back to skyrtuna.

So the structure is:

• Ég heng skyrtuna ...
• og set hana í skápinn

Even though English also repeats the object as it, learners should be careful not to drop hana.

Because Icelandic often omits the repeated subject in a coordinated sentence when it is clearly the same subject.

So:

• Ég heng skyrtuna á herðatréð og set hana í skápinn

means essentially:

• Ég heng skyrtuna á herðatréð og ég set hana í skápinn

Both are understandable, but leaving out the second ég is very natural because the subject is obviously still I.

Yes. It is a compound noun.

It is made from:

• herðar = shoulders
• tré = tree / wood

So historically it is something like shoulder-tree, which became the ordinary word for a clothes hanger.

Compound nouns are extremely common in Icelandic, so noticing them can help you remember vocabulary.