Ο γείτονας, ο σκύλος του οποίου γαβγίζει τη νύχτα, μου ζήτησε συγγνώμη σήμερα το πρωί.

The second ο introduces the noun σκύλος inside the relative clause:

Ο γείτονας, ο σκύλος του οποίου γαβγίζει τη νύχτα, ...

This relative clause literally means something like:

the neighbor, the dog of whom barks at night

So ο σκύλος is a complete noun phrase of its own, and it needs its own article.

In English, we usually just say whose dog barks at night, but Greek often spells this structure out more explicitly.

του οποίου means whose.

More literally, it means of whom or of which. In this sentence:

ο σκύλος του οποίου = whose dog

This is a more formal or structured way of expressing possession in a relative clause.

So:

• ο γείτονας = the neighbor
• ο σκύλος του οποίου = whose dog

Together:

Ο γείτονας, ο σκύλος του οποίου γαβγίζει τη νύχτα, ... = The neighbor, whose dog barks at night, ...

Because οποίου has to match the person being referred to, and it also has to be in the genitive to show possession.

Here, του οποίου refers back to ο γείτονας, which is:

• masculine
• singular

And because the meaning is possessive (whose dog = the dog of the neighbor), Greek uses the genitive.

So:

• ο οποίος = nominative masculine singular
• του οποίου = genitive masculine/neuter singular

That is why του οποίου is the correct form here.

Yes, in everyday speech, Greek often prefers που.

A more colloquial version would be:

Ο γείτονας, που ο σκύλος του γαβγίζει τη νύχτα, μου ζήτησε συγγνώμη σήμερα το πρωί.

This also means The neighbor, whose dog barks at night, apologized to me this morning.

The difference is mostly one of style:

• του οποίου = more formal, more written, more explicit
• που ... του = more common in everyday spoken Greek

So the original sentence is perfectly correct, but it sounds a bit more formal than casual speech.

σκύλος is in the nominative, because it is the subject of γαβγίζει.

In the relative clause:

ο σκύλος του οποίου γαβγίζει τη νύχτα

the dog is the one doing the barking, so ο σκύλος is the subject.

That is why it is nominative:

• ο σκύλος = the dog

If the dog were the object of a verb, it would be in a different case.

Greek often uses the accusative for expressions of time, especially when meaning during some period.

So:

• τη νύχτα = at night / during the night

This is very common in Greek. Similar examples:

• το πρωί = in the morning
• το βράδυ = in the evening
• την Κυριακή = on Sunday

So γαβγίζει τη νύχτα means barks at night.

In Greek, to apologize is commonly expressed with the phrase:

ζητώ συγγνώμη = literally I ask forgiveness = idiomatically I apologize

So:

• ζήτησε = he/she asked
• συγγνώμη = forgiveness / pardon / sorry
• μου = to me

Thus:

μου ζήτησε συγγνώμη = he apologized to me

This is a very natural Greek expression.

μου means to me.

It is the weak form of the first-person singular pronoun in the genitive, which Greek often uses where English uses an indirect object.

So:

μου ζήτησε συγγνώμη literally = he asked me forgiveness naturally = he apologized to me

Greek often uses these short pronouns before the verb:

• μου = to me / my
• σου = to you / your
• του / της = to him, to her / his, her

Here it is functioning as the indirect object: the apology was directed to me.

Because σήμερα το πρωί means this morning, while το πρωί by itself usually just means in the morning or morning in a more general sense.

So:

• το πρωί = in the morning
• σήμερα το πρωί = this morning

The sentence is specifying exactly when the apology happened: today in the morning, that is, this morning.

Because this is a non-restrictive relative clause: it adds extra information about the neighbor, rather than identifying which neighbor out of many.

Compare the idea in English:

• The neighbor, whose dog barks at night, apologized to me this morning.
• The clause whose dog barks at night is extra descriptive information.

If the speaker were trying to identify which neighbor they mean, the punctuation and phrasing could be different. But here, the clause is presented as additional information, so commas are used.

If you take out the extra descriptive part, the core sentence is:

Ο γείτονας μου ζήτησε συγγνώμη σήμερα το πρωί.

That means:

The neighbor apologized to me this morning.

So the structure is:

• Ο γείτονας = subject
• μου ζήτησε συγγνώμη = verb phrase
• σήμερα το πρωί = time expression

The relative clause in the middle just gives extra information about the neighbor:

ο σκύλος του οποίου γαβγίζει τη νύχτα = whose dog barks at night

Greek word order is more flexible than English word order, because case endings and articles make relationships clearer.

The given sentence is very natural, but some elements could move for emphasis. For example:

Σήμερα το πρωί ο γείτονας, ο σκύλος του οποίου γαβγίζει τη νύχτα, μου ζήτησε συγγνώμη.

This still means the same thing, but now σήμερα το πρωί is fronted for emphasis.

What is less flexible is the internal structure of the phrase ο σκύλος του οποίου, because that is a tightly connected unit meaning whose dog.

Yes, most naturally it does.

γαβγίζει τη νύχτα is in the present tense and suggests something habitual or repeated:

• barks at night
• not necessarily is barking right now

So the meaning is that the dog is known for barking at night, not that the barking is happening at the same moment as the apology.

This is similar to English:

• His dog barks at night = habitual
• His dog is barking tonight = happening now

Greek uses the simple present here in the same general way.