I børnehaven spørger læreren, om min datter har sin madkasse med.

This is because of normal Danish V2 word order in main clauses.

In Danish, the finite verb usually comes in the second position of the sentence. Here, I børnehaven has been moved to the front, so the verb spørger must come next:

• I børnehaven spørger læreren ...

If you started with the subject instead, you would get:

• Læreren spørger ...

Both are correct, but the first one puts extra focus on where this is happening.

Because Danish usually adds the definite article to the end of the noun.

• børnehave = kindergarten/daycare

• børnehaven = the kindergarten/daycare

• lærer = teacher
• læreren = the teacher

So the -en ending is doing the job that English the does before the noun.

Literally, yes: børne- relates to children, and have means garden. But in real usage, børnehave means kindergarten, preschool, or daycare, depending on context.

For an English speaker, it is important not to take it too literally. In Danish, børnehave usually refers to a childcare institution for young children, not necessarily exactly the same thing as US school kindergarten.

Here om means whether or if.

So:

• Læreren spørger, om min datter har sin madkasse med.
  = The teacher asks whether/if my daughter has her lunchbox with her.

Danish om can also mean about in other sentences, but not here. In this sentence it introduces an indirect yes/no question.

Because after om, you have a subordinate clause, and subordinate clauses in Danish usually have subject + verb order.

So:

• om min datter har sin madkasse med

This is different from the main clause, where Danish uses the V2 pattern. A very common thing for learners to notice is:

• main clause: often verb in second position
• subordinate clause: usually subject before verb

Because sin/sit/sine is the reflexive possessive in Danish. It is used when the owner is the subject of the same clause.

In the clause:

• om min datter har sin madkasse med

the subject is min datter, so sin means her own.

If you said hendes madkasse, it would usually suggest someone else's lunchbox belonging to her, or at least sound less clearly reflexive. So sin is the natural choice here.

Madkasse means lunchbox.

It is a compound noun:

• mad = food
• kasse = box

Danish forms compound nouns very freely, and they are normally written as one word. So where English often has a choice between one word, two words, or a hyphen, Danish usually prefers one combined word.

At have noget med means to have something with you or to bring something along.

So:

• har sin madkasse med
  means has her lunchbox with her / has brought her lunchbox

The word med belongs to the expression, but it is often placed later in the clause. This can feel a bit like an English phrasal verb or particle construction.

Yes. Danish present tense often covers both:

• asks
• is asking

So spørger can mean either, depending on context. Danish usually does not need a special continuous form like English is asking.

That is why a simple present form in Danish often sounds more flexible than the English simple present.

Because om introduces a subordinate clause.

In Danish, many writers put a comma before a subordinate clause like this:

• Læreren spørger, om ...

You may also see Danish sentences without that comma, depending on the comma style being used. So it is a normal place for a comma, but learners should know that punctuation conventions can vary.

Yes, that is also grammatical.

The difference is mostly about focus:

• I børnehaven spørger læreren ...
  puts the location first, so it highlights in the kindergarten/daycare

• Læreren spørger i børnehaven ...
  starts with the subject and sounds more neutral

So the original sentence is not using a different meaning, just a different emphasis.