Ertu til í að sauma hnappinn á skyrtuna mína aftur?

Because this is a yes/no question.

In Icelandic, yes/no questions usually put the verb first. So:

• Þú ert ... = You are ...
• Ertu ...? = Are you ...?

Ertu is a very common spoken-and-written contraction of ert þú:

• ert = are
• þú = you

So Ertu til í að ...? literally starts as Are you up for / willing to ...?

Til í að is an idiomatic expression meaning something like:

• willing to
• up for
• in the mood to
• okay with doing

So Ertu til í að sauma ...? means Are you willing/up for sewing ...?

A helpful thing to remember is that this expression does not translate word-for-word very neatly into natural English. You should learn vera til í að + infinitive as a chunk.

Examples:

• Ertu til í að hjálpa mér? = Are you willing to help me?
• Ég er til í það. = I’m up for that.

Here að is the infinitive marker, like English to before a verb.

So:

• að sauma = to sew

After til í, Icelandic normally uses að + infinitive:

• til í að sauma
• til í að hjálpa
• til í að koma

So in this sentence, að sauma simply means to sew.

Because after að, Icelandic uses the infinitive.

The verb is:

• að sauma = to sew

You are not saying you sew or she sews here. You are saying to sew, so the infinitive is required.

Compare:

• Ég sauma = I sew
• Hún saumar = she sews
• að sauma = to sew

Hnappinn is the accusative singular definite form of hnappur = button.

Breakdown:

• hnappur = a button
• hnappinn = the button

It is in the accusative because it is the direct object of sauma:

• you sew the button

So the verb sauma is acting on hnappinn.

This is a very common Icelandic case question.

The preposition á can take either:

• accusative for movement toward / onto / result
• dative for location / being on something

Here, the button is being sewn onto the shirt, so there is a sense of resulting placement. That is why Icelandic uses the accusative:

• á skyrtuna = onto the shirt

Compare:

• Hnappurinn er á skyrtunni. = The button is on the shirt.
  
  • location, so dative
• Sauma hnappinn á skyrtuna. = Sew the button onto the shirt.
  
  • movement/result, so accusative

In Icelandic, possessives like minn/mín/mitt often come after the noun, especially in ordinary speech and writing.

So:

• skyrtan mín = my shirt
• skyrtuna mína = my shirt, in the accusative definite form

Why mína?

Because it has to agree with skyrtuna, which is:

• feminine
• singular
• accusative
• definite

So the possessive also becomes:

• mína

This agreement is very important in Icelandic.

Aftur usually means again or back.

In this sentence it gives the sense of:

• sew the button on again
• sew the button back on

So it implies the button has probably come off before, and the speaker wants it reattached.

That is very natural Icelandic usage. A single word, aftur, often covers what English expresses with again or back on.

Yes, it is very natural and conversational.

Ertu til í að ...? is a common, friendly way to ask whether someone is willing to do something.

It is not extremely formal, but it is perfectly normal in everyday speech.

Depending on tone and context, English equivalents could be:

• Would you sew the button back onto my shirt?
• Would you mind sewing the button back onto my shirt?
• Are you up for sewing the button back onto my shirt?

So the Icelandic phrasing is casual, polite, and idiomatic.

Yes.

Both are possible:

• Ertu til í að ...
• Ert þú til í að ...

Ertu is the more compact and very common form.
Ert þú can sound a bit more deliberate or emphatic, depending on context.

For most everyday purposes, Ertu is the form learners will hear and see very often.

A few parts may be especially tricky for English speakers:

• Ertu
  
  • roughly EHRT-uh
• til í að
  
  • til EE ath
• sauma
  
  • roughly SOY-ma or SÖY-ma depending on how you hear it taught
• hnappinn
  
  • the hn at the beginning is unusual for English speakers; the h is not strongly pronounced like normal English h
• skyrtuna
  
  • y in Icelandic is not like English y; it is closer to an i-like vowel
• aftur
  
  • roughly AF-tur

The biggest pronunciation challenge is often not one single word, but getting used to Icelandic rhythm and vowel quality. Listening to native audio is especially helpful for phrases like Ertu til í að ...?

Yes, it could.

This sentence is structured as sew the button onto my shirt, which focuses on reattaching it.

If you wanted to describe the button on my shirt as a noun phrase, you might say something like:

• hnappurinn á skyrtunni minni = the button on my shirt

Notice the difference:

• á skyrtuna mína = onto my shirt
  
  • accusative, result/movement
• á skyrtunni minni = on my shirt
  
  • dative, location

So the original sentence is chosen because it fits the action of sewing the button back onto the shirt.