Stuur de doos terug en vraag het na.

Because this is an imperative sentence: it gives an instruction or command.

In Dutch, the imperative usually uses just the verb stem:

• sturen → Stuur!
• vragen → Vraag!

So:

• Stuur de doos terug
• Vraag het na

You do not normally need to say jij. Dutch works like English here:

• Send the box back
• Ask/check it

The subject is understood.

The verb is terugsturen, meaning to send back / return. This is a separable verb.

With separable verbs in Dutch:

• in the infinitive, it is one word: terugsturen
• in a main clause with a finite verb, it splits: stuur ... terug

So:

• Ik stuur de doos terug.
• Stuur de doos terug.
• We moeten de doos terugsturen.

This is a very common Dutch pattern.

This is another separable verb: navragen.

It means something like:

• to check by asking
• to ask about something in order to confirm it
• to follow up on something

Just like terugsturen, it splits in a main clause:

• infinitive: navragen
• main clause: vraag ... na

So:

• Ik vraag het na.
• Vraag het na.
• Je moet het navragen.

The na is not functioning as ordinary after here. It is the separated particle of the verb navragen.

Probably not.

In this sentence, het usually means it in the sense of the matter / the issue / the information that needs to be checked.

So vraag het na means something like:

• check it
• verify it
• ask about it to make sure

It does not naturally refer to de doos, because doos is a de-word, not a het-word.

So the sentence is more like:

• Send the box back and check/confirm it.

That second part is about the situation, not the physical box itself.

Because doos is a de-word in Dutch:

• de doos = the box

This is something you mostly have to learn with the noun. Dutch nouns take either de or het.

A useful extra point:

• singular: de doos
• plural: de dozen
• diminutive: het doosje

So even though doos itself takes de, the diminutive doosje takes het, because Dutch diminutives are always het-words.

With separable verbs, the usual order in a main clause is:

finite verb + object + separated particle

So:

• Ik vraag het na.
• Vraag het na.
• Hij stuurt de doos terug.

That is why het comes before na.

Vraag na het is not correct here.

You can think of it a bit like an English phrasal verb:

• check it out not
• check out it

The comparison is not perfect, but the placement of a short pronoun before the particle is similar.

Usually no, or at least it would sound incomplete here.

Navragen normally needs an object: you ask/check something.

So Dutch usually wants:

• Vraag het na.
• Vraag dat na.
• Vraag de details na.

Without an object, the listener may wonder: Ask/check what?

Using het is very common when the thing being checked is understood from the context.

It is fairly direct, because it uses the imperative.

That does not automatically make it rude. It can sound perfectly normal in:

• instructions
• manuals
• workplace directions
• informal conversation

But if you want to sound more polite, Dutch often uses a different structure, for example:

• Kunt u de doos terugsturen en het navragen?
• Wilt u de doos terugsturen en het navragen?

A softer informal version might add even:

• Stuur de doos terug en vraag het even na.

Here even makes the instruction sound a bit lighter, like just in English.

Plain vragen just means to ask.

Navragen means more specifically:

• to ask around
• to check something by asking
• to confirm something
• to follow up

So:

• Ik vraag het. = I ask it / I ask the question
• Ik vraag het na. = I check/confirm it by asking

That extra idea of verifying or following up is why navragen is used here.