Hvorug okkar trúir því að hurðarklukkan hafi hringt svona seint.

Hvorug is the feminine singular nominative form of hvorugur, meaning neither.

So hvorug okkar means neither of us, but the exact form of hvorugur changes for gender. In this sentence, hvorug suggests that the two people referred to are feminine.

Compare:

• hvorugur okkar = neither of us (if masculine / mixed, depending on context)
• hvorug okkar = neither of us (feminine)

This is one of those details English does not show, but Icelandic does.

Because hvorug means neither one, and grammatically that is singular.

Even though two people are involved overall, Icelandic treats neither as a singular subject:

• Hvorug okkar trúir ... = neither of us believes ...

So the singular verb trúir is exactly what you would expect.

Okkar is the genitive plural form of við (we/us).

After hvorug(ur), Icelandic commonly uses the pattern:

• hvorug(ur) + genitive = neither of ...

So:

• okkar = of us
• hvorug okkar = neither of us

This is similar to English of us, but Icelandic expresses that of relationship with the genitive case instead of a separate word.

Because the verb trúa normally takes a dative object.

So in a structure like this, Icelandic often says:

• trúa því að ...
  literally: believe it/that, that ...

Here því is the dative pronoun required by trúa, and að introduces the following clause.

So the structure is:

• trúir því = believes that / believes it
• að hurðarklukkan hafi hringt ... = that the doorbell rang / has rung ...

This is very natural in Icelandic, even though English does not need an extra pronoun there.

No. In this sentence, því and að are doing two separate jobs.

• því is the dative pronoun connected to trúir
• að introduces the subordinate clause

So this is not the conjunction því að meaning because.

The structure is:

• trúir því = believes that / believes it
• að ... = that ...

That is different from:

• því að = because

Because hafi is the subjunctive form of hafa, and Icelandic often uses the subjunctive in subordinate clauses involving belief, doubt, uncertainty, or non-factual statements.

Here the idea is not presented as an established fact. The sentence is about what neither person believes. That makes the subordinate clause a good environment for the subjunctive:

• að hurðarklukkan hafi hringt = that the doorbell has rung / rang

So hafi signals something like claimed / supposed / not accepted as fact.

If you used hefur, the clause would sound more like a straightforward factual statement.

Hafi is the present subjunctive of the verb hafa (to have).

In the perfect tense, Icelandic uses hafa + a verbal form like hringt:

• hefur hringt = has rung
• hafi hringt = has rung, but in the subjunctive

So hafi hringt is the subjunctive perfect form of hringja in this clause.

Because after hafa, Icelandic uses the supine form of the verb, which often looks the same as the neuter singular past participle.

For hringja (to ring), that form is hringt.

So:

• hefur hringt
• hafi hringt

This is just the normal perfect construction. English also uses a special form here:

• has rung

Hurðarklukkan is a compound noun:

• hurð = door
• klukka = bell
• hurðarklukka = doorbell
• hurðarklukkan = the doorbell

The -ar- part is the linking/genitive element from hurð inside the compound.

Then the definite article is attached to the end:

• hurðarklukka = a doorbell
• hurðarklukkan = the doorbell

This is very typical Icelandic noun-building.

Because Icelandic is referring to a specific doorbell, just as English would usually say the doorbell in this context.

Instead of a separate word like the, Icelandic usually adds the article to the noun:

• hurðarklukka = a doorbell
• hurðarklukkan = the doorbell

So the definiteness is built directly into the word.

Because after að, you get a subordinate clause, and Icelandic subordinate clauses do not usually follow the same verb-second pattern as main clauses.

So this is normal subordinate-clause order:

• að hurðarklukkan hafi hringt

with:

• hurðarklukkan = subject
• hafi = finite verb
• hringt = supine / main verb form

In a main clause, Icelandic often puts the finite verb in second position. But after að, the structure is different and more like what you see here.

Seint means late, and svona adds the idea of like this / so / this.

So svona seint means something like:

• this late
• so late
• at such a late hour

It makes the statement feel more vivid than just seint by itself.

Compare:

• seint = late
• svona seint = this late / so late

In standard Icelandic, trúa því að ... is the normal and idiomatic pattern.

A learner should generally treat því as part of how this construction works:

• trúi því að ...
• trúir því að ...
• trúðum því að ...

So for practical purposes, no: you should learn this as trúa því að ..., not just trúa að ....

Yes. Icelandic can use hringja for something like a bell or a doorbell ringing.

So:

• hurðarklukkan hringir = the doorbell rings / is ringing
• hurðarklukkan hefur hringt = the doorbell has rung
• hurðarklukkan hafi hringt = that the doorbell has rung, in the subjunctive setting used here

That is a perfectly natural use of the verb.