Ég get fyrirgefið henni, þó að ég hafi verið reið í gær.

After geta in Icelandic, the next verb is normally in the supine form, not the infinitive.

For this verb:

• að fyrirgefa = infinitive
• fyrirgefið = supine

So:

• Ég get fyrirgefið henni = I can forgive her

This is a very common thing that surprises English speakers, because fyrirgefið looks like a past-participle-type form, but here it is just the normal form used after geta.

Because fyrirgefa takes the dative case.

The pattern is:

• fyrirgefa einhverjum = to forgive someone

So with the pronoun hún:

• hún = nominative
• hana = accusative
• henni = dative

That means henni is the correct form here.

Þó að means although, even though, or though.

It introduces a subordinate clause that adds a contrast:

• Ég get fyrirgefið henni = I can forgive her
• þó að ég hafi verið reið í gær = even though I was angry yesterday

So the whole sentence has a concessive meaning: one fact is true despite another fact.

No. Here að is simply part of the fixed conjunction þó að.

So in this sentence:

• þó að = although / even though

It is not the infinitive marker here.

That is different from something like:

• að fyrirgefa = to forgive

Also notice that after geta, you do not use the infinitive marker að.

Yes. Þótt is a very common equivalent of þó að.

So these are both possible in the same kind of sentence:

• Ég get fyrirgefið henni, þó að ég hafi verið reið í gær.
• Ég get fyrirgefið henni, þótt ég hafi verið reið í gær.

The meaning is essentially the same. Learners should just recognize both.

Because þó að commonly triggers the subjunctive in Icelandic.

Here:

• hafa = to have
• hafi = present subjunctive of hafa

So:

• þó að ég hafi verið... is a normal subjunctive clause after þó að

This is one of the most useful patterns to learn:

• þó að / þótt
  • subjunctive

English usually does not show this kind of mood difference so clearly, which is why it can feel unfamiliar.

That is a very good question. Icelandic tense usage does not match English tense usage exactly.

Even though hafi verið looks like have been, in this kind of clause Icelandic often uses this form naturally where English would simply use the simple past:

• þó að ég hafi verið reið í gær
• natural English: even though I was angry yesterday

So you should not translate the tense word-for-word. The important idea is that the anger was a prior state, and English expresses that more naturally here with was.

Because the adjective agrees with the speaker’s grammatical gender.

Here:

• reið = feminine
• reiður = masculine

So this sentence is spoken by a female speaker:

• ég hafi verið reið

A male speaker would say:

• Ég get fyrirgefið henni, þó að ég hafi verið reiður í gær.

It modifies the subordinate clause:

• ég hafi verið reið í gær = I was angry yesterday

It does not go with get fyrirgefið.

So the sentence means:

• I can forgive her now,
• even though I was angry yesterday.

Yes.

You can say:

• Þó að ég hafi verið reið í gær, get ég fyrirgefið henni.

This is very useful to notice because the main clause then shows normal Icelandic V2 word order:

• get ég
• not ég get

So when something else comes first, the finite verb usually comes before the subject in the main clause.

For a learner, the safest rule is: usually yes.

In standard Icelandic, þó að / þótt very often appears with the subjunctive, especially in this kind of concessive sentence. You may sometimes come across other patterns, but if you are learning the structure, þó að + subjunctive is the right one to master first.