Prebivalište i boravište su u zahtevu.

In official or legal Serbian, they are not the same:

• prebivalište = your permanent registered residence
• boravište = your temporary residence or place where you are currently staying

So on forms and in bureaucracy, these are standard technical terms, not just two casual synonyms for address.

They are both singular.

Even though they end in -e, they are singular neuter nouns.
Their plural forms would be:

• prebivališta
• boravišta

So in this sentence, you have two singular nouns joined by i.

They are neuter nouns.

A lot of Serbian nouns ending in -šte are neuter. In the dictionary, you would normally see them in this same basic form:

• prebivalište
• boravište

They are in the nominative singular.

That is because they are the subject of the sentence: they are the things being talked about. In other words, they are the things that are in the application.

Because the subject is compound: prebivalište i boravište = residence and temporary residence.

Each noun is singular, but together they make a plural subject, so Serbian uses the plural form of to be:

• je = is
• su = are

Compare:

• Prebivalište je u zahtevu. = only one subject
• Prebivalište i boravište su u zahtevu. = two subjects joined by i

Because the preposition u here means in and shows location, so it requires the locative case.

The basic form is:

• zahtev = request / application

In the locative singular, it becomes:

• u zahtevu = in the application / in the request

So zahtevu is simply the correct case form after u in this sentence.

No. u can take either:

• locative for location: u zahtevu = in the application
• accusative for movement into something: for example u kuću = into the house

In this sentence there is no movement, only location, so locative is used.

Because Serbian does not have articles like a and the.

So words like prebivalište, boravište, and zahtevu can often be understood as:

• the residence
• the temporary residence
• the application

The exact meaning depends on context, not on an article.

Yes. Serbian word order is fairly flexible.

This sentence could also be:

• U zahtevu su prebivalište i boravište.

That version puts more focus on where they are.
The original sentence is perfectly correct too; it just starts with the things being discussed.

In administrative Serbian, zahtev often means:

• request
• application
• sometimes form, depending on context

If this is a bureaucratic document, application is often the most natural English equivalent. Serbian uses the same word in many official situations, so the best translation depends on what kind of document it is.

A rough pronunciation guide:

• prebivalište ≈ preh-bee-vah-LEE-shte
• boravište ≈ boh-rah-VEE-shte
• zahtevu ≈ ZAH-hteh-voo

A useful detail: š sounds like sh in ship.

So the endings -šte sound roughly like -shte.