Инспектор явно занят.

In Russian, the verb to be is usually omitted in the present tense.

So:

• Инспектор занят = The inspector is busy
• literally: Inspector busy

This is completely normal in Russian.
You would only use forms of быть in the past or future:

• Инспектор был занят = The inspector was busy
• Инспектор будет занят = The inspector will be busy

Инспектор is in the nominative case because it is the subject of the sentence.

It is the person being described as busy.

Here the basic structure is:

• Инспектор = subject
• явно = adverb
• занят = predicate adjective / short-form adjective

Занят is the short form of the adjective, and short forms are commonly used after the subject to mean a temporary state or condition.

So:

• Инспектор занят = The inspector is busy

The full form is different:

• занятый инспектор = a busy inspector

That means the adjective is directly modifying the noun.

So compare:

• Инспектор занят = The inspector is busy.
• Занятый инспектор = The busy inspector.

For this sentence, занят is the correct form because it describes the inspector’s current state.

Yes. Занят agrees with инспектор in gender and number.

Because инспектор is masculine singular, the short form is:

• masculine: занят
• feminine: занята
• neuter: занято
• plural: заняты

Examples:

• Инспектор занят = The inspector is busy.
• Инспекторша занята = The female inspector is busy.
• Окно занято = The window is occupied.
• Все инспекторы заняты = All the inspectors are busy.

Явно is an adverb meaning something like:

• clearly
• obviously
• plainly
• sometimes apparently, depending on context

In this sentence, it means that the inspector is obviously / clearly busy.

So Инспектор явно занят has the sense of:

• The inspector is clearly busy
• The inspector is obviously busy

The standard stress is:

• инспе́ктор
• я́вно
• заня́т

So the whole sentence is pronounced:

инспе́ктор я́вно заня́т

Stress is important in Russian, because it is not always predictable.

Yes, Russian word order is flexible, but changing it affects emphasis.

Neutral order:

• Инспектор явно занят.

Other possible orders:

• Явно инспектор занят.
• Инспектор занят явно.
• Занят инспектор, явно. (more marked, unusual in normal speech)

The most natural and neutral version is the original:

• Инспектор явно занят.

That order simply states the fact without special emphasis.

Yes, but the meaning changes.

• явно = clearly / obviously
• очень = very

So:

• Инспектор явно занят = The inspector is obviously busy.
• Инспектор очень занят = The inspector is very busy.

Both are correct, but they express different ideas:

• явно comments on how noticeable the situation is
• очень comments on degree

No. Занят / занята / занято / заняты can also mean occupied for places, seats, lines, rooms, and so on.

For example:

• Место занято. = The seat is occupied.
• Телефон занят. = The line is busy.
• Комната занята. = The room is occupied.

With a person, it usually means busy.
With an object or place, it often means occupied.

Grammatically, инспектор is a masculine noun. In many contexts, it can refer to a person of either sex, especially as a job title, but the grammar is still usually masculine unless the speaker chooses another form.

So in standard grammar:

• инспектор занят = masculine agreement

If the speaker specifically wants feminine agreement for a female person, usage can vary depending on style, region, and context. In many formal situations, masculine professional titles are still common.

For a learner, the important point is:

• инспектор is grammatically masculine
• so here it takes занят

It is neutral standard Russian.

• инспектор is a normal noun
• явно is common and natural
• занят is standard grammar

So the sentence could appear in everyday speech, writing, or narration without sounding strange.