Hún þurrkar skiptiborðið eftir að hún skiptir um bleyju.

Because there are two clauses here:

• Hún þurrkar skiptiborðið
• eftir að hún skiptir um bleyju

Each clause has its own subject, and in Icelandic it is normal to state hún again in the second clause. English does the same here: She wipes the changing table after she changes a diaper.

Þurrkar is the 3rd person singular present tense of þurrka, meaning to wipe or to dry.

So:

• ég þurrka = I wipe
• hún þurrkar = she wipes

In this sentence, it matches hún: Hún þurrkar = She wipes.

That -ið is the suffixed definite article, which is very common in Icelandic. Instead of using a separate word like English the, Icelandic often adds the article to the end of the noun.

• skiptiborð = changing table
• skiptiborðið = the changing table

So the word already includes the meaning of the.

Here skiptiborðið is the direct object of þurrkar, so it is in the accusative.

However, skiptiborð is a neuter noun, and for many neuter nouns the nominative and accusative singular look the same. So even though it is accusative here, the form is still skiptiborðið.

That can feel strange at first, but it is very normal in Icelandic.

Yes. Skiptiborð is a compound noun:

• skipti relates to changing
• borð = table

So skiptiborð is literally something like changing-table, that is, a table used for changing a baby.

Icelandic makes a lot of compound nouns, much like English does.

Because eftir að introduces a full clause:

• eftir að hún skiptir um bleyju = after she changes a diaper

By contrast, eftir by itself is used before a noun phrase:

• eftir bleyjuskiptin = after the diaper change

So:

• eftir + noun
• eftir að + clause

That is a very useful pattern to remember.

This is an idiomatic verb expression: skipta um means to change/replace something.

So:

• skipta on its own can have meanings like divide, share, exchange, or switch
• skipta um specifically means change in the sense of replacing one thing with another

So hún skiptir um bleyju means she changes a diaper.

The um is not optional here if you want this meaning.

Bleyju is the accusative singular of the feminine noun bleyja.

That happens because um in this expression takes the accusative:

• bleyja = a diaper (nominative)
• bleyju = a diaper (accusative)
• bleyjan = the diaper (nominative definite)
• bleyjuna = the diaper (accusative definite)

So in this sentence:

• um bleyju = a diaper

It is indefinite because the sentence is talking about changing a diaper, not a specific already-identified diaper. If you meant a specific one, you could say um bleyjuna.

Because Icelandic verbs do not all follow the same present-tense pattern.

In this sentence:

• þurrka → þurrkar
• skipta → skiptir

Both are normal 3rd person singular present forms, but they belong to different conjugation patterns. This is something learners gradually get used to: you often need to learn a verb together with its present-tense forms.

So the difference is not caused by the sentence structure; it is just part of how those verbs are conjugated.

After eftir að, the clause here has normal subordinate-clause order:

• eftir að hún skiptir um bleyju

That is:

• conjunction
• subject
• verb
• rest of clause

If you put the eftir að clause first, the main clause then shows normal Icelandic verb-second behavior:

• Eftir að hún skiptir um bleyju þurrkar hún skiptiborðið.

Notice that in the main clause, þurrkar comes before hún when the subordinate clause is placed first.

Most naturally, this sounds like a habitual or general action:

• She wipes the changing table after she changes a diaper.

In other words, this is something she does as a routine.

But the Icelandic present tense can also be used in narration or for actions viewed as happening in the present situation. Context decides that. Without extra context, the habitual reading is the most natural one here.