Ég set glósubókina á borðið og skrifa í henni.

Because glósubókina means the notebook, while glósubók means a notebook.

Icelandic usually adds the definite article to the end of the noun:

• glósubók = notebook
• glósubókin = the notebook

In this sentence, the noun is also in the accusative case because it is the direct object of set. That gives:

• glósubókina = the notebook (accusative singular)

So Ég set glósubókina... means I put the notebook...

Setja is the infinitive: to put / to place.

In the sentence, you need the present tense form for I:

• að setja = to put
• ég set = I put

So:

• Ég set glósubókina... = I put the notebook...

This is just the normal conjugated verb form after ég.

Yes — and that is something that often confuses learners.

With skrifa, the 1st person singular present happens to look exactly like the infinitive form without að:

• að skrifa = to write
• ég skrifa = I write

So in this sentence, skrifa is a finite verb meaning I write, not an infinitive.

Because á can take different cases depending on whether there is movement or location.

Here, the notebook is being moved onto the table, so Icelandic uses accusative:

• á borðið = onto the table

If the notebook were already there and you were just describing its position, Icelandic would use dative:

• á borðinu = on the table

A useful rule:

• motion toward a place → often accusative
• being in a place → often dative

So:

• Ég set glósubókina á borðið = I put the notebook onto the table

Because here í means in in the sense of location, not movement into.

When you are writing in the notebook, the writing is taking place in it, so Icelandic uses the dative:

• í henni = in it

If there were movement into something, Icelandic would often use the accusative instead:

• í hana = into it (feminine)

So compare:

• Ég skrifa í henni. = I write in it.
• Ég set blað í hana. = I put a sheet into it.

Because Icelandic pronouns follow grammatical gender, not natural gender.

Glósubók is a feminine noun, so when you refer back to it, you use a feminine pronoun.

Here the form must also be dative because of í with location, so the pronoun becomes:

• hún = she / it (feminine nominative)
• hana = her / it (feminine accusative)
• henni = her / it (feminine dative)

So í henni literally looks like in her, but in English we translate it as in it, because it refers to the notebook.

It refers to glósubókina — the notebook.

So:

• Ég set glósubókina á borðið og skrifa í henni.
• ...and write in it

Even though borðið is also nearby in the sentence, henni must refer to glósubókina, because:

1. the meaning makes sense with the notebook, and
2. henni is feminine, while borð is neuter

So the grammar helps you identify the referent.

Because Icelandic, like English, can leave out the repeated subject when two verbs share the same subject and are linked by og.

So this is natural:

• Ég set glósubókina á borðið og skrifa í henni.

It means:

• I put the notebook on the table and write in it.

You could repeat ég, but it is usually unnecessary:

• Ég set glósubókina á borðið og ég skrifa í henni.

That sounds more emphatic or stylistically heavier.

Yes, absolutely.

You could say:

• Ég set glósubókina á borðið og skrifa í glósubókinni.

That is grammatical, and it means essentially the same thing. But it sounds more repetitive because the noun has just been mentioned.

Notice the form:

• glósubókinni = in the notebook

That is the dative singular definite form, because í here expresses location.

So both are possible:

• í henni = in it
• í glósubókinni = in the notebook

Yes. In Icelandic, skrifa í is the normal expression when you write in a book, notebook, journal, and similar things.

For example:

• skrifa í bók = write in a book
• skrifa í dagbók = write in a diary
• skrifa í glósubók = write in a notebook

So skrifa í henni is completely natural here.

A learner often wants to map the whole sentence, so here it is:

• Ég — nominative, because it is the subject
• glósubókina — accusative, because it is the direct object of set
• á borðið — accusative after á, because it shows movement onto the table
• í henni — dative after í, because it shows location in the notebook

So the sentence is a nice example of how Icelandic case works with both verbs and prepositions.