Ο βοηθός σκηνοθέτη ελέγχει το πρόγραμμα προβών.

Because σκηνοθέτη is the genitive singular form of σκηνοθέτης. Greek often uses the genitive after a noun like βοηθός to mean assistant of / assistant to someone.
So:

• ο σκηνοθέτης = the director
• ο βοηθός σκηνοθέτη = the assistant of the director / assistant director

If you used σκηνοθέτης, that would be the nominative form, which would not fit this structure.

Greek often leaves out the article in phrases like this when the second noun functions almost like a label or category. So βοηθός σκηνοθέτη can sound a bit like a professional title, especially in theatre or film language.

Compare:

• ο βοηθός σκηνοθέτη = assistant director / director’s assistant as a role
• ο βοηθός του σκηνοθέτη = the assistant of the director, with stronger emphasis on a specific director

Both are possible, but the version without του is very natural in this kind of context.

Very often, yes. In a theatre or film context, βοηθός σκηνοθέτη is commonly understood as assistant director. Greek frequently uses a noun + genitive structure for job titles or role descriptions.

So although the phrase can literally be understood as the director’s assistant, in context it may be better thought of as a set role.

Πρόγραμμα can mean several related things: program, schedule, plan, or timetable, depending on context.

In το πρόγραμμα προβών, the most natural English meaning is usually the rehearsal schedule or the rehearsal timetable, not a formal printed program. The noun is broad, and context tells you which English word fits best.

Προβών is the genitive plural of πρόβα, which means rehearsal.

So:

• η πρόβα = the rehearsal
• οι πρόβες = the rehearsals
• πρόγραμμα προβών = schedule of rehearsals

Greek often uses the genitive where English uses of or an attributive noun. So instead of saying something like rehearsal schedule directly, Greek says schedule of rehearsals.

Without the article, προβών sounds more general: a schedule of rehearsals or the rehearsal schedule as a type of thing.

If you say το πρόγραμμα των προβών, that points more clearly to a specific set of rehearsals already known in the conversation: the schedule of the rehearsals.

So:

• το πρόγραμμα προβών = more general / label-like
• το πρόγραμμα των προβών = more specific and definite

Ελέγχει is 3rd person singular, present tense of ελέγχω.

That means it corresponds to:

• he checks
• she checks
• it checks
• or, depending on context, is checking

Greek present tense can cover both simple present and continuous present, so ελέγχει can mean either checks or is checking.

The subject is ο βοηθός σκηνοθέτη, and the object is το πρόγραμμα προβών.

A few things help you see that:

• ο marks βοηθός as nominative singular masculine, which strongly suggests the subject
• ελέγχει is a transitive verb, so it can take a direct object
• το πρόγραμμα προβών comes after the verb and makes sense as the thing being checked

One small complication is that το πρόγραμμα has the same form in the nominative and accusative, because it is neuter. So here, word order and meaning help confirm that it is the object.

Because Greek articles change for gender, number, and case.

Here:

• ο = masculine nominative singular
• το = neuter nominative/accusative singular

So:

• ο βοηθός = the assistant
• το πρόγραμμα = the schedule/program

Also, βοηθός can be masculine or feminine depending on the article:

• ο βοηθός = a male assistant
• η βοηθός = a female assistant

Yes. Greek word order is more flexible than English word order because case and articles help show the grammatical relationships.

The sentence as given is a neutral subject–verb–object order:

Ο βοηθός σκηνοθέτη ελέγχει το πρόγραμμα προβών.

But Greek could also say things like:

• Το πρόγραμμα προβών ελέγχει ο βοηθός σκηνοθέτη.
• Ελέγχει το πρόγραμμα προβών ο βοηθός σκηνοθέτη.

These versions would sound different in emphasis, but the basic meaning would stay the same.

The written accent marks show the stressed syllables:

• βοηθός
• σκηνοθέτη
• ελέγχει
• πρόγραμμα
• προβών

A rough pronunciation guide is:

o voi-THOS ski-no-THE-ti e-LENG-hi to PRO-gra-ma pro-VON

A few useful sound notes:

• οι and η sound like ee
• χ is not English ch; it is the Greek χ sound
• the stress matters, so it is important to notice the accent marks rather than flattening the words evenly