Jeg håber, at printeren ikke går i stykker igen.

At is the conjunction that.

So Jeg håber, at printeren ikke går i stykker igen literally has the structure I hope that the printer does not break again.

In everyday Danish, at is often optional after verbs like håber, so you will also hear:

Jeg håber printeren ikke går i stykker igen.

Both are correct. Including at can sound a little clearer or slightly more formal.

Because Danish usually makes nouns definite by adding the definite article to the end of the noun.

• en printer = a printer
• printeren = the printer

So printeren is the normal way to say the printer.

You can say den printer, but that usually has a more demonstrative meaning, closer to that printer or that particular printer.

Danish does not use do-support the way English does.

In English, you say:

• The printer does not break
• Does the printer break?

In Danish, the main verb itself carries the tense, and you just add ikke for negation:

• printeren går ikke i stykker = the printer does not break / is not breaking down

So there is no extra helper verb equivalent to English do/does in this kind of sentence.

Because this is a subordinate clause introduced by at.

In Danish subordinate clauses, sentence adverbs like ikke usually come before the finite verb.

So:

• at printeren ikke går i stykker igen

But in a main clause, the finite verb usually comes earlier, before ikke:

• Printeren går ikke i stykker igen.

This is a very important Danish word-order pattern.

Yes. Gå i stykker is a very common expression meaning to break, to break down, or to stop working.

Literally, it is something like go into pieces, but you should learn it as a whole expression.

Examples:

• Min telefon gik i stykker. = My phone broke.
• Bilen er gået i stykker. = The car has broken down.

It is especially common for machines, objects, and things that stop functioning.

The infinitive is at gå.

In this sentence, the verb is in the present tense:

• går = present tense of gå

But because it is part of the expression gå i stykker, you should think of the whole unit:

• at gå i stykker = to break / to break down
• går i stykker = breaks / is breaking down

Danish often uses the present tense for future meaning when the context already makes the time clear.

Here, Jeg håber ... igen clearly refers to something that might happen later, so present tense is natural:

• Jeg håber, at printeren ikke går i stykker igen.

This works much like English in sentences such as:

• I hope it doesn’t rain tomorrow.

So Danish does not need a separate future form here.

Igen means again, and putting it at the end is the most neutral and natural position here.

So:

• printeren ikke går i stykker igen

That final position sounds very normal.

You may sometimes see adverbs moved for emphasis in Danish, but in this sentence, the end position is the standard choice.

No, that word order is not correct after at.

After at, Danish uses subordinate-clause word order, where ikke comes before the finite verb:

• Jeg håber, at printeren ikke går i stykker igen.

If it were a main clause, then går ikke would be correct:

• Printeren går ikke i stykker igen.

So this is one of those places where Danish word order changes depending on whether the clause is main or subordinate.

Yes, but the meaning may change slightly.

For example:

• bliver ødelagt = gets damaged / gets ruined
• holder op med at virke = stops working

But går i stykker is one of the most natural everyday ways to talk about a printer, phone, car, or other object breaking or failing.

So for this sentence, går i stykker is an excellent and very idiomatic choice.