Η ταυτοπροσωπία και η φωτογραφία διαβατηρίου είναι απαραίτητες.

Because in Greek, when two definite nouns are joined by και (and), it is very normal to repeat the definite article with each noun.

So:

• η ταυτοπροσωπία
• η φωτογραφία

This is the standard way to say the proof of identity and the passport photo.

Also, Greek uses the definite article more often than English does, especially in formal or bureaucratic language. So even if English might say proof of identity and passport photo are required, Greek naturally says η ... και η ....

No. Ταυτοπροσωπία does not mean identity card.

It is a formal word meaning something like:

• proof/verification that the person is who they say they are
• identification of the person
• confirmation of identity

So in official contexts, ταυτοπροσωπία refers to identity verification, not to the physical document ID card.

If you wanted ID card, Greek would usually use ταυτότητα.

Because διαβατηρίου is in the genitive case.

Greek often uses the genitive to show a relationship like:

• φωτογραφία διαβατηρίου = passport photo
• literally something like photo of/for a passport

The base form is:

• το διαβατήριο = the passport

Its genitive singular is:

• του διαβατηρίου

Here, the noun appears as διαβατηρίου without the article because it functions as a dependent noun after φωτογραφία, giving the type of photo.

Because διαβατηρίου is not the main noun of the phrase. The main noun is η φωτογραφία.

The word διαβατηρίου is attached to it and specifies what kind of photo it is:

• η φωτογραφία διαβατηρίου = the passport photo

This is different from:

• η φωτογραφία του διαβατηρίου

That would usually mean the photo of the passport itself, as in a picture showing the passport document.

So:

• φωτογραφία διαβατηρίου = passport photo
• φωτογραφία του διαβατηρίου = photo of the passport

In the present tense, είναι is the same form for both:

• he/she/it is
• they are

So by itself, είναι does not tell you singular vs. plural.

You understand it from the rest of the sentence. Here, the subject is made of two nouns joined by και:

• η ταυτοπροσωπία και η φωτογραφία διαβατηρίου

That makes the subject plural in meaning, so είναι here means are.

The adjective απαραίτητες also confirms this, because it is plural.

Because it agrees with the two nouns it describes.

Both nouns are feminine singular:

• η ταυτοπροσωπία
• η φωτογραφία

When two singular nouns are joined by και, together they form a plural idea. Since both nouns are feminine, the adjective becomes feminine plural:

• singular feminine: απαραίτητη
• plural feminine: απαραίτητες

So:

• Η φωτογραφία είναι απαραίτητη. = The photo is necessary.
• Η ταυτοπροσωπία και η φωτογραφία ... είναι απαραίτητες. = The proof of identity and the photo are necessary.

They are in the nominative case, because they are the subject of the sentence.

You can think of the structure as:

• Η ταυτοπροσωπία και η φωτογραφία διαβατηρίου = subject
• είναι = verb
• απαραίτητες = predicate adjective

So the main subject nouns are nominative, while διαβατηρίου is genitive because it depends on φωτογραφία.

Yes. Greek word order is more flexible than English word order.

The version you have is a neutral, standard sentence:

• Η ταυτοπροσωπία και η φωτογραφία διαβατηρίου είναι απαραίτητες.

But Greek could also say:

• Απαραίτητες είναι η ταυτοπροσωπία και η φωτογραφία διαβατηρίου.

This puts more emphasis on απαραίτητες.

The original order is probably the most natural one for a learner to remember, especially in formal written Greek.

Yes, it sounds fairly formal and bureaucratic.

That is mainly because of words like:

• ταυτοπροσωπία
• απαραίτητες

This is the kind of wording you might see on:

• application instructions
• government forms
• official notices
• consulate or visa requirements

In everyday speech, people might use simpler wording, but this sentence is very natural in official contexts.