Zalepi kutiju da bi ostala zatvorena.

Yes. Zalepi is the 2nd person singular imperative of the perfective verb zalepiti.

Here it means something like:

• seal
• stick shut
• glue/tape shut

The exact English verb depends on context. With a box, it often suggests seal the box or tape the box shut.

It is also perfective, which fits a one-time, completed action very well: do this action and achieve the result.

Because kutiju is in the accusative singular.

• kutija = nominative, the basic dictionary form
• kutiju = accusative, used here because the box is the direct object of zalepi

So the structure is:

• Zalepi = seal/stick
• kutiju = the box

In other words, what should be sealed? → the box

Da bi introduces a purpose clause. In English, it often corresponds to:

• so that
• in order that
• so as to make sure that

So:

• Zalepi kutiju da bi ostala zatvorena.
• Seal the box so that it stays closed.

This is a very common Serbian way to express purpose.

That is a very common learner question.

Here ostala is not a simple past tense. It is part of the conditional construction after da bi:

• da bi + l-participle

So:

• da bi ostala = so that it would remain / stay

Serbian often uses this structure where English uses so that or in order that.

The verb behind it is ostati = to remain / to stay.

Because they agree with kutija, which is a feminine singular noun.

• kutija = feminine singular
• ostala = feminine singular
• zatvorena = feminine singular

Even though kutija appears as kutiju in the main clause, it is still the same noun, and in the subordinate clause it is understood as the thing that would remain closed.

If the noun were masculine, the forms would change. For example:

• paket → ostao zatvoren

Because Serbian often omits subject pronouns when they are understood from context.

In English, you need it:

• Seal the box so that it stays closed.

In Serbian, the verb form and context already make it clear that the understood subject is kutija.

So Serbian naturally says:

• da bi ostala zatvorena

without needing an explicit ona.

Because zatvorena is a predicate adjective, not a direct object.

This is the key idea:

• In the main clause, kutiju is the object of zalepi.
• In the da bi clause, the box is the understood subject of ostala.

So inside that subordinate clause, the adjective agrees with the subject and stays in the nominative feminine singular:

• ostala zatvorena = remained closed

It is not:

• ostala zatvorenu

because that would treat the adjective like an accusative object complement, which is not what is happening here.

Yes, that is also natural.

There is a slight nuance:

• da ostane zatvorena = very common, straightforward, natural
• da bi ostala zatvorena = a bit more explicitly purpose-oriented, often felt as slightly more formal or deliberate

Both can mean:

• Seal the box so that it stays closed.

In everyday speech, many speakers would probably use da ostane zatvorena very comfortably.

Because that is the normal order with this kind of predicate in Serbian.

• ostala zatvorena = remained closed

The adjective describes the state resulting from the verb ostati.

This is similar to English patterns like:

• stay calm
• remain silent
• remain closed

So the Serbian order here is very natural.

This is about aspect.

• zalepiti = perfective
• lepiti = imperfective

In this sentence, zalepiti fits better because the speaker wants a single completed action with a clear result:

• Seal the box

If you used lepi, it would sound more like:

• be sealing / keep sealing / do the action in progress

That is usually not what is meant here.

So zalepi is the natural choice for a one-time instruction.

Not necessarily.

The basic idea is make something stick / attach / seal by sticking. Depending on context, it can mean:

• glue
• stick
• tape
• seal shut

With a box, English often prefers:

• tape the box shut
• seal the box

So the Serbian verb is a bit broader than just literal glue.