Ang lalaki ang direktor.

Because this is an identification sentence: one noun phrase is being identified as the other.

Filipino can say the equivalent of X is Y without a separate word for is in the present tense. In this sentence, both lalaki and direktor are full noun phrases, and both are marked with ang.

So the structure is basically:

Ang lalaki = ang direktor

A good learner-friendly way to understand it is: the man = the director.

It is not stated. In Filipino, the present-tense copula is/are/am is often omitted.

So:

Ang lalaki ang direktor
literally feels like The man, the director

but it means:

The man is the director.

This is very normal in Filipino.

Not exactly.

Ang often overlaps with English the, especially in simple beginner sentences, but it is not a perfect one-to-one translation. It is better to think of ang as a marker for a focused/topic/common noun phrase.

In this sentence, translating ang as the works well:

• ang lalaki = the man
• ang direktor = the director

But in other sentences, ang does not behave exactly like the English article the.

The difference is mainly definiteness and what kind of statement you are making.

• Ang lalaki ang direktor = The man is the director
  This identifies the man as a specific director.

• Direktor ang lalaki = The man is a director or The man is director
  This sounds more like describing his role, job, or class, not necessarily identifying him as the specific director.

So if you mean He is the director, the version with ang direktor is the better match.

Yes. It is grammatical, but the emphasis changes.

• Ang lalaki ang direktor = The man is the director
• Ang direktor ang lalaki = The director is the man

Both can describe the same real-world situation, but they start from different points. The first starts with the man; the second starts with the director.

English does this too:

• The man is the director
• The director is the man

Same basic identity, different presentation.

Yes. You can say:

Ang lalaki ay ang direktor.

That is grammatical and can sound more formal, careful, or written. In everyday speech, Filipino often leaves out ay, so:

Ang lalaki ang direktor

is perfectly normal.

A helpful rule of thumb:

• without ay = often more conversational
• with ay = often more formal or more explicitly structured

Because lalaki and direktor are common nouns, not personal names.

Use si for names of people:

• Si Maria ang direktor.
• Si Juan ang lalaki.
  (less natural in meaning, but grammatically it shows the pattern)

Use ang for common nouns:

• Ang lalaki
• Ang direktor

So in your sentence, ang is correct.

Yes, especially if you are identifying who the director is.

For example, if several people are being discussed and you want to clarify which one is the director, Ang lalaki ang direktor makes sense.

But if you simply want to say what the man's profession is, Filipino often prefers something like:

Direktor ang lalaki.
= The man is a director.

So the given sentence sounds most natural when the meaning is specifically The man is the director.

It is a loanword, ultimately from Spanish and also very close to English director. Loanwords are very common in Filipino, and they behave like normal Filipino nouns in sentences.

So even though direktor is borrowed, it still works naturally with Filipino markers like ang:

• ang direktor
• ang mga direktor

This is completely normal.