Geturðu fyrirgefið mér?

Because Icelandic yes/no questions normally put the finite verb first.

• Statement: Þú getur fyrirgefið mér. = You can forgive me.
• Question: Geturðu fyrirgefið mér? = Can you forgive me?

So this is normal Icelandic question word order.

-ðu is the pronoun þú attached to the verb.

So:

• Geturðu = Getur þú
• literally can you

This kind of attachment is very common in Icelandic, especially in everyday speech and writing.

Yes. That is grammatical.

But Geturðu fyrirgefið mér? is usually the more natural, everyday form. Writing the pronoun separately can sound a little more careful or slightly more emphatic.

After geta, Icelandic normally uses the supine form of the main verb (in Icelandic grammar, sagnbót), not the plain infinitive.

For fyrirgefa, that form is fyrirgefið.

So the pattern is:

• geta fyrirgefið
• geta hjálpað
• geta komið
• geta séð

This is one of the things English speakers often have to get used to, because English uses the plain verb after can.

No, not in this sentence.

It is true that fyrirgefið! can also be the plural imperative, meaning forgive! when speaking to more than one person. But in Geturðu fyrirgefið mér?, it is not an imperative. Here it is the verb form used after geta.

The context makes that clear:

• Geturðu ... ? = a question
• fyrirgefið = the main verb form after geta

Because fyrirgefa takes the person in the dative case.

So:

• mér = to me / me (dative)
• mig = me (accusative)

With this verb, Icelandic says fyrirgefa einhverjum = forgive someone, so mér is the correct form.

If you also mention the thing being forgiven, that thing is often separate:

• Geturðu fyrirgefið mér þetta? = Can you forgive me this? / Can you forgive me for this?

No. Nothing is missing.

After geta, you do not use að. The main verb follows directly:

• geta fyrirgefið
• geta hjálpað
• geta komið

So Geturðu fyrirgefið mér? is complete as it stands.

Yes, if you literally mean Can you forgive me?

It sounds like a real request for forgiveness.

Compare:

• Geturðu fyrirgefið mér? = Can you forgive me?
• Fyrirgefðu mér. = Forgive me.
• Afsakaðu mig. = Excuse me / Sorry.

So this sentence is more emotional and personal than a casual sorry.

þú is singular, but Icelandic does not normally use a special formal singular pronoun like French vous or German Sie. Using þú is standard in most situations.

If you are speaking to more than one person, use þið:

• Getið þið fyrirgefið mér? = Can you all forgive me?

Use the conditional form of geta:

• Gætirðu fyrirgefið mér?

This sounds softer, more tentative, or more polite than Geturðu fyrirgefið mér?.