Prima di entrare in ufficio, chiedo il permesso alla segretaria.

Because prima di + infinitive is the normal Italian pattern for before doing something.

• prima di entrare = before entering
• prima di parlare = before speaking
• prima di uscire = before going out

So di is required here before the infinitive entrare.

If the subject changes, Italian often uses a full clause instead:

• Prima di entrare, chiedo il permesso. = before entering, I ask permission.
• Prima che lui entri, gli parlo. = before he enters, I speak to him.

Because after prima di, Italian uses the infinitive when the subject is the same as the subject of the main verb.

In this sentence, the person who asks permission is also the person who enters. So Italian uses:

• Prima di entrare...

not a conjugated verb.

If the two actions have different subjects, Italian usually switches to prima che + subjunctive:

• Prima che la segretaria entri, aspetto.

Both are possible, but they are not exactly the same.

• in ufficio is more general and idiomatic. It often means into the office / into the workplace / to the office area.
• nell’ufficio points more clearly to a specific office/room.

So:

• entrare in ufficio = enter the office / go into the office
• entrare nell’ufficio del direttore = enter the director’s office

In your sentence, in ufficio sounds natural because it refers generally to entering the office.

Because with some common location expressions, Italian often uses the noun without an article, especially in set phrases.

Examples:

• sono in ufficio
• vado in ufficio
• entrare in ufficio

This is similar to how English sometimes says at school, at home, in prison, depending on the expression.

If you want to emphasize one particular office, the article can appear:

• entro nell’ufficio della segretaria
• sono nell’ufficio del direttore

In standard Italian, the usual expression is chiedere il permesso.

So Italian treats permesso here as a noun with the definite article:

• chiedere il permesso a qualcuno
• chiedo il permesso alla segretaria

This is the standard way to say ask permission or ask for permission.

You may sometimes hear shorter expressions in speech, but for learners, chiedere il permesso is the safest and most natural pattern.

Because the person you ask is an indirect object in Italian, and indirect objects are introduced by a.

The basic pattern is:

• chiedere qualcosa a qualcuno = ask someone for something

So:

• chiedo = I ask
• il permesso = permission
• alla segretaria = to the secretary

alla = a + la

Other examples:

• chiedo un consiglio a Marco
• chiedo informazioni alla receptionist

Not in this sentence.

La segretaria would be a direct object, but here the secretary is the person to whom the request is made, so Italian needs a:

• correct: chiedo il permesso alla segretaria
• not correct here: chiedo il permesso la segretaria

A useful comparison:

• Vedo la segretaria = I see the secretary
• Chiedo il permesso alla segretaria = I ask the secretary for permission

The present tense in Italian can describe:

• a habitual action
• a general truth
• a present-time action
• even a narrative style

So chiedo here most naturally sounds like:

• Before entering the office, I ask the secretary for permission
• in other words, something I normally do

If you wanted a one-time past action, you would use a past tense:

• Prima di entrare in ufficio, ho chiesto il permesso alla segretaria.

Yes. That word order is completely natural.

Both versions are correct:

• Prima di entrare in ufficio, chiedo il permesso alla segretaria.
• Chiedo il permesso alla segretaria prima di entrare in ufficio.

The difference is mainly emphasis.

• Starting with Prima di entrare in ufficio highlights the time sequence first.
• Starting with Chiedo il permesso puts the main action first.

The comma after the opening phrase is normal and helpful.

Not always. It depends on context.

Segretaria can mean:

• secretary
• sometimes receptionist
• an administrative office worker

In English, secretary and receptionist are often more clearly separated than in Italian. So in some contexts, segretaria may refer to the person at the front desk or the person managing access to an office.

Related, but not exactly the same structure.

• chiedere il permesso a qualcuno = ask someone for permission
• chiedere il permesso di fare qualcosa = ask permission to do something

Examples:

• Chiedo il permesso alla segretaria.
• Chiedo il permesso di entrare.
• Chiedo alla segretaria il permesso di entrare.

All of these are possible, depending on how much detail you want to include.

Usually, chiedo is better here.

Both chiedere and domandare can mean to ask, but chiedere is the more natural verb for:

• asking for something
• requesting permission
• asking a favor

So:

• chiedere il permesso = standard and natural
• domandare il permesso = possible, but less common in everyday use

A good rule:

• chiedere is the safer choice for requests
• domandare is often more about asking a question

It is:

time expression + main verb + direct object + indirect object

More specifically:

• Prima di entrare in ufficio = time phrase
• chiedo = main verb
• il permesso = direct object
• alla segretaria = indirect object

So the sentence is built very logically:

1. first, the timing is given
2. then the main action
3. then what is asked
4. then the person asked