Η κτηνιατρική εξέταση αρχίζει τώρα.

Η is the definite article meaning the. It is used here because εξέταση is a feminine singular noun, so the article must match it.

• η = the, for feminine singular nouns in the nominative
• εξέταση = examination / exam

So Η εξέταση means the examination.

Here κτηνιατρική is an adjective meaning veterinary. It describes εξέταση.

Because εξέταση is feminine singular, the adjective also appears in the feminine singular form:

• κτηνιατρικός = veterinary, masculine
• κτηνιατρική = veterinary, feminine
• κτηνιατρικό = veterinary, neuter

So κτηνιατρική εξέταση means veterinary examination.

A learner may also notice that κτηνιατρική can sometimes be used as a noun by itself, meaning veterinary medicine or the veterinary field, but in this sentence it is clearly an adjective.

In Greek, adjectives can often come before or after the noun. Here, putting κτηνιατρική before εξέταση is completely natural.

So these are both possible in many contexts:

• η κτηνιατρική εξέταση
• η εξέταση η κτηνιατρική or η εξέταση κτηνιατρική would not be the normal equivalent here

For this sentence, η κτηνιατρική εξέταση is the standard, natural phrasing for the veterinary examination.

Αρχίζει is the 3rd person singular present tense of αρχίζω, meaning to begin or to start.

So:

• αρχίζω = I begin / I start
• αρχίζεις = you begin
• αρχίζει = he/she/it begins

Here the subject is η κτηνιατρική εξέταση, which is grammatically third person singular, so Greek uses αρχίζει:

• Η κτηνιατρική εξέταση αρχίζει = The veterinary examination begins

Greek often does not use subject pronouns when the verb ending already shows who or what the subject is. This is called a pro-drop language.

In English, you need it begins.
In Greek, αρχίζει already means begins, and the subject is already stated as η κτηνιατρική εξέταση, so no extra pronoun is needed.

A sentence like Η κτηνιατρική εξέταση αυτή αρχίζει τώρα would mean something different and add emphasis or a different structure. In the original sentence, the plain version is best.

Τώρα means now. It tells you when the action happens.

So:

• Η κτηνιατρική εξέταση αρχίζει τώρα = The veterinary examination begins now

It is an adverb of time.

No. Greek word order is fairly flexible, and τώρα can move depending on emphasis.

For example:

• Η κτηνιατρική εξέταση αρχίζει τώρα. = neutral, very natural
• Τώρα αρχίζει η κτηνιατρική εξέταση. = stronger focus on now
• Η κτηνιατρική εξέταση τώρα αρχίζει. = possible, but less neutral and more marked

So the original sentence is a normal, natural way to say it.

The subject phrase Η κτηνιατρική εξέταση is in the nominative case.

Why?

Because it is the thing that begins.

You can see the agreement clearly:

• η = feminine singular nominative article
• κτηνιατρική = feminine singular nominative adjective
• εξέταση = feminine singular nominative noun

They all match because they belong to the same noun phrase.

Yes, depending on context. Greek present tense often covers both:

• begins / starts
• is beginning / is starting

So Η κτηνιατρική εξέταση αρχίζει τώρα could be understood as:

• The veterinary examination begins now
• The veterinary examination is starting now

The best English version depends on the situation and the style of translation.

A simple pronunciation guide would be:

ee ktee-nee-ah-TREE-kee ek-SEH-ta-see ar-KHEE-zee TO-ra

A few helpful notes:

• η is pronounced like ee
• κτ at the start of κτηνιατρική may feel unusual for English speakers, but both consonants are pronounced
• χ in αρχίζει is not like English ch in chair; it is a harsher sound, like the h sound in some pronunciations of German Bach, though before ι it is softer
• the written accent mark shows the stressed syllable:
  • κτηνιατρική
  • εξέταση
  • αρχίζει
  • τώρα

Yes, it works as a normal set phrase meaning veterinary examination, veterinary exam, or veterinary check-up, depending on context.

Greek often builds this kind of phrase with:

• article + adjective + noun

So η κτηνιατρική εξέταση is a very natural way to name this kind of examination.