Fæ ég að opna gluggann?

Here fæ is the 1st person singular present tense of fá, which often means to get or to receive.

But in the pattern fá að + infinitive, it means to be allowed to or to get to do something.

So:

• fæ ég að opna gluggann? = May I open the window? / Can I open the window?
• literally, something like Do I get to open the window?

Because this is a question.

In Icelandic, yes/no questions often have verb-first word order:

• statement: Ég fæ að opna gluggann. = I may / get to open the window.
• question: Fæ ég að opna gluggann? = May I open the window?

This is similar to English word order in questions:

• I can open the window.
• Can I open the window?

In this sentence, að is the infinitive marker, like English to in to open.

So:

• opna = open
• að opna = to open

After fá in this meaning, Icelandic normally uses the pattern:

• fá að + infinitive

Examples:

• Ég fæ að fara. = I’m allowed to go.
• Færðu að koma inn? = May you come in? / Are you allowed to come in?

So að is required here.

Opna is the infinitive of the verb að opna, meaning to open.

That is why it appears after að.

Compare:

• að opna = to open
• ég opna = I open
• hann opnar = he opens

In your sentence, the speaker is asking for permission to open the window, so the infinitive is used.

Gluggann means the window.

The base noun is:

• gluggi = window

But here the noun is the direct object of opna, so it must be in the accusative case. Also, it is definite: the window.

So the forms are:

• gluggi = a window / window (nominative singular)
• gluggann = the window (accusative singular with the definite article)

This is a very common Icelandic pattern: the definite article is attached to the end of the noun.

In Icelandic, the is usually not a separate word. Instead, it is added to the end of the noun as a suffix.

So:

• glugga / gluggi = window (depending on case)
• gluggann = the window

The ending -ann here is part of the definite form in the accusative singular for this masculine noun.

So opna gluggann literally means open-the-window.

Usually permission.

Fæ ég að opna gluggann? most naturally means:

• May I open the window?
• Am I allowed to open the window?

It is not mainly about physical ability. If you wanted to ask more about ability, Icelandic would more naturally use geta:

• Get ég opnað gluggann? = Can I open the window? / Am I able to open the window?

So:

• fá að = permission / being allowed
• geta = ability / being able

Yes, absolutely.

Má ég opna gluggann? is also a very common way to say May I open the window?

A useful difference is:

• Má ég opna gluggann? = May I open the window?
  This is the most direct and standard permission question.
• Fæ ég að opna gluggann? = Do I get to / Am I allowed to open the window?
  This can sound slightly more like asking whether permission will be given.

In many everyday situations, both are natural.

A rough learner-friendly pronunciation is:

• Fæ ≈ fye
• ég ≈ yeg or yeh-g
• að ≈ often like ath or a softer að sound, depending on speech
• opna ≈ OP-na
• gluggann ≈ GLUG-gan

Very roughly as a whole:

fye yeg ath OP-na GLUG-gan?

A few helpful notes:

• æ is pronounced somewhat like the i in English fine
• g in ég is not a hard English g
• double gg in gluggann is pronounced with a short preceding vowel

If you want, I can also give you a more precise pronunciation guide.

Ég is in the nominative case, because it is the subject of the sentence.

Even though the sentence begins with the verb, ég is still the person doing the action:

• Fæ ég... ? = May I... ?

So:

• ég = I (nominative)
• subject of the verb fæ

The sentence follows this pattern:

• fá
  • subject
    • að + infinitive
      • object

More specifically:

• Fæ = present tense of fá
• ég = subject
• að opna = infinitive phrase, to open
• gluggann = direct object, the window

So the structure is:

Fæ ég að opna gluggann?
= Am I allowed to open the window?

This is a very useful pattern to learn:

• Fæ ég að fara? = May I go?
• Fæ ég að spyrja? = May I ask?
• Fær hann að koma? = Is he allowed to come?

As a statement, it would be:

Ég fæ að opna gluggann.

That means:

• I may open the window
• I am allowed to open the window
• I get to open the window

The only big change is word order:

• question: Fæ ég að opna gluggann?
• statement: Ég fæ að opna gluggann.

This verb-first vs. subject-first contrast is very important in Icelandic.