Κτηνιατρικές εξετάσεις φαίνονται στο ηλεκτρονικό βιβλιάριο υγείας και στον ιατρικό φάκελο.

Because Greek often leaves out the article when talking about something in a general or indefinite way, especially in the plural.

So:

• Κτηνιατρικές εξετάσεις = veterinary examinations/tests in a general sense
• Οι κτηνιατρικές εξετάσεις would sound more like the veterinary examinations

In this sentence, the noun phrase is being presented generally, not as a specifically identified set.

It is in the nominative plural feminine.

You can tell it is the subject because:

• εξετάσεις is a nominative plural form
• the verb φαίνονται is 3rd person plural
• the verb agrees with εξετάσεις

So the structure is:

• Κτηνιατρικές εξετάσεις = subject
• φαίνονται = verb
• στο ηλεκτρονικό βιβλιάριο υγείας και στον ιατρικό φάκελο = location phrases

Because its subject, Κτηνιατρικές εξετάσεις, is plural.

• εξέταση = examination/test
• εξετάσεις = examinations/tests

So Greek, like English, makes the verb agree with the subject:

• singular: Η εξέταση φαίνεται
• plural: Οι εξετάσεις φαίνονται

Here φαίνονται means something like:

• are visible
• appear
• show up
• can be seen

It comes from φαίνομαι, which is a very common verb in Greek.

In this sentence, it does not mean that someone is actively showing them. It means that the examinations are visible / appear in those records.

Because φαίνομαι is one of those Greek verbs that commonly appears in middle/passive-style forms, even when the meaning is not a straightforward passive.

So although φαίνονται ends in -ονται, here it functions naturally as the normal present form of φαίνομαι:

• φαίνομαι = I appear / I seem / I am visible
• φαίνονται = they appear / they are visible

So it is best to learn φαίνομαι as its own verb, not just as a passive of another verb.

These are contractions of σε + article:

• στο = σε το
• στον = σε τον

They are different because the nouns have different genders:

• βιβλιάριο is neuter → το βιβλιάριο → στο βιβλιάριο
• φάκελος is masculine → τον φάκελο → στον φάκελο

So the sentence uses the correct form depending on the noun that follows.

Because Greek articles and adjectives must match the noun in gender, number, and case.

Here:

• βιβλιάριο is neuter singular

  • το ηλεκτρονικό βιβλιάριο
  • after σε: στο ηλεκτρονικό βιβλιάριο

• φάκελο is masculine singular accusative

  • τον ιατρικό φάκελο
  • after σε: στον ιατρικό φάκελο

So the adjective endings change to match the noun:

• ηλεκτρονικό matches neuter βιβλιάριο
• ιατρικό matches masculine accusative φάκελο

Because it means of health.

Greek often uses the genitive where English uses a noun before another noun:

• βιβλιάριο υγείας = health booklet literally booklet of health
• φάκελος ασθενούς = patient file literally file of patient

So υγείας is the genitive singular of υγεία.

Βιβλιάριο is a small booklet or record book.

It comes from βιβλίο (book) with the suffix -άριο, which often gives the sense of a small or specialized book/document.

In real-life Greek, βιβλιάριο υγείας is a standard expression for a health record booklet or health booklet.

Yes. Ιατρικός φάκελος means medical file, medical record, or medical chart, depending on context.

• ιατρικός = medical
• φάκελος = folder, file

So στον ιατρικό φάκελο means in the medical record/file.

In Modern Greek, σε normally takes the accusative.

That is why we get:

• σε το βιβλιάριο → στο βιβλιάριο
• σε τον φάκελο → στον φάκελο

Even if older Greek used other patterns, in Modern Greek you should generally think:

• σε + accusative

Yes.

• Κτηνιατρικές εξετάσεις φαίνονται...
• Οι κτηνιατρικές εξετάσεις φαίνονται...

Both are possible, but they are not exactly identical in feel.

Without the article, the phrase is more general or indefinite.
With the article, it points more clearly to a known or specific set of examinations.

So the version without οι is natural if the sentence is giving general information.

No, Greek word order is fairly flexible.

This sentence uses a very natural order:

• subject → verb → location phrases

But Greek could rearrange it for emphasis, for example:

• Στο ηλεκτρονικό βιβλιάριο υγείας και στον ιατρικό φάκελο φαίνονται κτηνιατρικές εξετάσεις.

That would put more emphasis on where the examinations appear.

So the original order is normal, but not the only possible one.

Here are the dictionary-style singular forms:

• κτηνιατρική εξέταση = veterinary examination/test
• ηλεκτρονικό βιβλιάριο = electronic booklet
• υγεία = health
• ιατρικός φάκελος = medical file/record

And in the sentence they appear as:

• κτηνιατρικές εξετάσεις = nominative plural
• βιβλιάριο = accusative singular neuter after σε
• υγείας = genitive singular
• φάκελο = accusative singular masculine after σε

Greek often avoids repetition when the structure is clear.

So:

• στο ηλεκτρονικό βιβλιάριο υγείας και στον ιατρικό φάκελο

means:

• in the electronic health booklet and in the medical file

The preposition is effectively understood with both items, but it is repeated in contracted form because the article changes:

• στο for the neuter noun
• στον for the masculine noun

That makes the phrase sound natural and grammatically complete.