Poslovnica banke je danas otvorena.

Because banke is in the genitive singular.

In poslovnica banke, Serbian expresses possession or close relationship with the genitive, so it literally means branch of the bank.

• banka = bank
• banke = of the bank

This is very common in Serbian:

• vrata kuće = the door of the house
• direktor škole = the director of the school
• poslovnica banke = the bank’s branch / the branch of the bank

Because poslovnica is the subject of the sentence.

The sentence is saying something about the branch office: it is open today.

So:

• poslovnica = nominative singular, because it is the thing being described
• otvorena agrees with it
• je is the verb is

Structure:

• Poslovnica banke = the bank branch
• je otvorena = is open
• danas = today

Je is the 3rd person singular present tense of biti (to be).

So here it means is.

• ja sam = I am
• ti si = you are
• on/ona/ono je = he/she/it is

In this sentence:

• Poslovnica banke je danas otvorena. = The bank branch is open today.

In normal standard Serbian, no: you usually keep je.

So:

• Poslovnica banke je danas otvorena. = correct standard sentence

Leaving it out would sound incomplete in ordinary prose and speech.

You may sometimes see omission in:

• headlines
• notes
• very compressed style

But for a learner, it is safest to treat je as necessary here.

Because it agrees with poslovnica, which is a feminine singular noun.

In Serbian, adjectives and adjective-like forms agree with the noun in gender, number, and case.

Since poslovnica is feminine singular nominative, the form is:

• otvoren = masculine singular
• otvorena = feminine singular
• otvoreno = neuter singular
• otvoreni / otvorene / otvorena = plural, depending on gender pattern

Compare:

• Restoran je otvoren. = The restaurant is open.
• Poslovnica je otvorena. = The branch is open.
• Pismo je otvoreno. = The letter is open / opened.

It can relate to both ideas, depending on context, because it comes from the verb otvoriti (to open) and can function like a passive participle or adjective.

In this sentence, the natural meaning is open in the sense of not closed / available for customers.

So:

• Poslovnica banke je danas otvorena. most naturally means
• The bank branch is open today.

If the context were different, otvorena could sometimes feel closer to opened, but here open is the right interpretation.

Yes, Serbian word order is fairly flexible, but the position changes the focus or emphasis.

Your sentence:

• Poslovnica banke je danas otvorena.

This is a neutral, natural sentence.

You could also say:

• Danas je poslovnica banke otvorena.
• Poslovnica banke danas je otvorena.
• Poslovnica banke je otvorena danas.

These all basically mean the same thing, but the rhythm and emphasis differ.

One important point: je is a clitic, and clitics in Serbian tend to stand in second position in the sentence or clause. That is why when danas is moved to the front, you usually get:

• Danas je poslovnica banke otvorena.

not

• Danas poslovnica banke je otvorena. in neutral standard usage

A clitic is a short, unstressed word that cannot stand freely in just any position. In Serbian, forms like je, sam, si, se, ga, mi, and so on often follow the second-position rule.

That means they usually come after the first stressed element of the clause.

Examples:

• Danas je otvorena.
• Poslovnica je otvorena.
• Ta poslovnica je otvorena.

So in your sentence, je is where you would expect it to be.

This is one of the most important word-order habits in Serbian.

Because Serbian does not have articles like English a/an and the.

So poslovnica banke can mean:

• a bank branch
• the bank branch

The exact meaning depends on context.

In this sentence, English usually chooses the:

• The bank branch is open today.

But Serbian does not mark that with an article.

Yes, depending on context.

Because Serbian has no articles, the sentence itself does not explicitly distinguish:

• A bank branch is open today from
• The bank branch is open today

Usually the situation tells you which one is intended.

If the speaker and listener already know which branch is being discussed, English would normally use the.

Here banke is genitive singular, not plural.

The base noun is:

• banka = bank

Its genitive singular is:

• banke = of the bank

This form happens to look the same as the nominative plural of many feminine nouns, which can confuse learners. But here the meaning and structure make it clearly genitive singular.

So:

• poslovnica banke = branch of the bank

not:

• branch banks

Yes, it is a very natural way.

Serbian often uses the pattern:

• noun + genitive noun

So poslovnica banke is a normal expression for bank branch or branch of a bank.

You may also hear filijala banke, depending on region or style, but poslovnica banke is perfectly standard and common.

You negate je with nije.

So:

• Poslovnica banke danas nije otvorena. or
• Poslovnica banke nije danas otvorena. if you want unusual emphasis
• most naturally: Poslovnica banke danas nije otvorena.

This means:

• The bank branch is not open today.

Notice that nije is the negative form of je.

Yes, but that changes the meaning.

• Banka je danas otvorena. = The bank is open today.
• Poslovnica banke je danas otvorena. = The bank branch is open today.

The second sentence is more specific: it talks about a particular branch office, not the bank as an institution in general.

It is historically related to a passive participle of otvoriti, but in sentences like this it functions very much like an adjective in the predicate.

For a learner, the most useful thing is:

• it agrees like an adjective
• it is used after je
• it describes the state of the noun

So in practice, you can think of je otvorena here as is open.

They would change the form of otvoren to match the noun.

Examples:

• Restoran je danas otvoren.
  The restaurant is open today.
  (restoran = masculine)

• Odeljenje je danas otvoreno.
  The department/ward is open today.
  (odeljenje = neuter)

• Poslovnica banke je danas otvorena.
  The bank branch is open today.
  (poslovnica = feminine)

This is a good way to practice agreement.

Yes. That is grammatically fine and very close to the Serbian structure.

But in more natural English, people would usually say:

• The bank branch is open today.

So:

• The branch office of the bank is open today = more literal
• The bank branch is open today = more natural English

The Serbian itself is completely normal.