Nem tudom, hogy a könyvtár még mindig nyitva van-e.

-e is a question particle used in embedded yes/no questions.

So in this sentence:

• Nem tudom = I don’t know
• hogy a könyvtár még mindig nyitva van-e = whether the library is still open

The -e shows that the clause is a whether/if clause, not a plain statement.

Compare:

• A könyvtár nyitva van. = The library is open.
• Nem tudom, hogy a könyvtár nyitva van-e. = I don’t know whether the library is open.

In English, we use if or whether. In Hungarian, this idea is often shown by -e on the verb.

They do different jobs.

• hogy introduces a subordinate clause. It often means that, but in some contexts it can introduce a clause that English would translate with whether.
• -e specifically marks the embedded clause as a yes/no question.

So:

• Nem tudom, hogy... = I don’t know that / I don’t know whether... depending on the rest of the clause
• ...van-e makes it clear that the clause means whether it is...

In practice, in a sentence like this, hogy + -e works together very naturally.

Yes, very often it can.

You can say:

• Nem tudom, hogy a könyvtár még mindig nyitva van-e.
• Nem tudom, a könyvtár még mindig nyitva van-e.

Both are natural. The version with hogy may sound a little more explicit or formal in some contexts, but both are standard.

Nyitva van is the normal way to say is open in the sense of being in an open state.

• nyitva = open in a state/result sense
• van = is

So nyitva van literally means something like is in an open state.

This pattern is very common in Hungarian:

• zárva van = is closed
• kinyitva van = is opened / has been opened
• becsukva van = is shut

Nyitott also exists, but it is not always interchangeable with nyitva.
For example:

• nyitott ablak = an open window
• A bolt nyitva van. = The shop is open.

For places like shops, offices, libraries, etc., nyitva van is the usual expression.

In neutral Hungarian, the predicate often appears as:

• nyitva van = is open

That same structure stays together when -e is added:

• nyitva van-e

Hungarian word order is flexible, so other orders can exist in certain contexts, especially for emphasis, but nyitva van-e is the normal neutral form here.

Még mindig means still.

The two words work together:

• még = originally still / yet / more
• mindig = always

But together, még mindig is a fixed expression meaning still.

So:

• a könyvtár még mindig nyitva van = the library is still open

You usually learn még mindig as a chunk.

A könyvtár means the library.

Hungarian uses the definite article a / az much like English the.
Since the sentence refers to a specific library, Hungarian normally uses the article:

• a könyvtár = the library

Without the article, könyvtár would sound more like library as a general concept, and it would not fit this sentence naturally.

Because Hungarian uses tud differently from English know in this structure.

• tudok = I can / I am able to
• tudom = I know it

Here, tud takes a definite object: the whole subordinate clause, understood as a specific thing that is known or not known. That is why the verb appears in the definite conjugation:

• tudom = I know it
• nem tudom = I don’t know it / I don’t know

This is very common with clause objects:

• Tudom, hogy igaz. = I know that it’s true.
• Nem tudom, hogy jön-e. = I don’t know whether he’s coming.

Yes, it is very natural.

A very common Hungarian pattern is:

• Nem tudom, hogy ...-e.

For example:

• Nem tudom, hogy igaz-e. = I don’t know whether it’s true.
• Nem tudom, hogy otthon van-e. = I don’t know whether he/she is at home.
• Nem tudom, hogy a könyvtár még mindig nyitva van-e.

You may also hear:

• Nem tudom, ...-e
• Nem tudom, vajon ...-e

where vajon adds a sense like I wonder whether.

Yes. A direct question would be:

• A könyvtár még mindig nyitva van? = Is the library still open?

That is a normal yes/no question.
But in your sentence, it is not a direct question. It is an embedded question after I don’t know, so Hungarian uses -e:

• ...nyitva van-e

That is one of the main differences between direct and embedded yes/no questions in Hungarian.

In a neutral reading, Hungarian stress usually falls on the first syllable of each word, but sentence-level emphasis depends on what is new or important.

A neutral version is something like:

• Nem tudom, hogy a könyvtár még mindig nyitva van-e.

If you want to emphasize still, you might stress még mindig more strongly.
If you want to emphasize the library rather than some other place, you might stress a könyvtár more.

Hungarian uses word order and intonation for emphasis, so the neutral order here does not strongly contrast anything.

Yes, Hungarian word order is flexible, but different orders can sound more marked or emphatic.

Neutral:

• Nem tudom, hogy a könyvtár még mindig nyitva van-e.

Possible alternative:

• Nem tudom, hogy még mindig nyitva van-e a könyvtár.

This can also sound natural. It may place the focus a bit differently, depending on context.

So the sentence you have is a normal, standard version, but not the only possible one.

-e attaches to the element that carries the question marking, often the verb.

Here that is:

• van-e

In other sentences, it may attach to another verb form:

• jön-e = whether he/she is coming
• igaz-e = whether it is true
• lehet-e = whether it is possible

So it is not specifically tied to van; it is tied to the predicate of the embedded yes/no question.

Yes. Hungarian normally separates the main clause from the subordinate clause with a comma:

• Nem tudom, hogy a könyvtár még mindig nyitva van-e.

That comma is standard because hogy a könyvtár... is a subordinate clause.

Even if hogy is omitted, the comma usually remains:

• Nem tudom, a könyvtár még mindig nyitva van-e.

Not always.

English if has two major uses:

1. whether/if in reported or embedded questions

   • I don’t know if he is here.
   • Hungarian often uses -e: Nem tudom, itt van-e.

2. if in conditional sentences

   • If it rains, we’ll stay home.
   • Hungarian does not use -e for that. It uses words like ha:
   • Ha esik, otthon maradunk.

So in your sentence, English if/whether corresponds to the embedded-question pattern with -e, not the conditional ha pattern.