Om kaninen er vågen eller træt, vil min datter altid børste dens pels og give den en godbid.

Here om means whether, not about.

Danish om has several meanings, and one of them is to introduce an indirect yes/no question or alternative. In this sentence, it sets up two possibilities:

• kaninen er vågen
• kaninen er træt

So Om kaninen er vågen eller træt means Whether the rabbit is awake or tired.

Because om kaninen er vågen eller træt is a subordinate clause, not a direct question.

In Danish:

• Direct question: Er kaninen vågen? = Is the rabbit awake?
• Subordinate clause: om kaninen er vågen = whether the rabbit is awake

After om, Danish normally uses regular subordinate-clause word order: subject + verb.

This is because Danish is a V2 language in main clauses. That means the finite verb usually comes in the second position.

Here, the whole opening clause Om kaninen er vågen eller træt takes the first position. So the finite verb of the main clause, vil, has to come next:

• Om kaninen er vågen eller træt, vil min datter altid ...

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

• Min datter vil altid børste dens pels ...

So the word order changes because something other than the subject has been placed first.

Because altid is placed after the finite verb and subject, but before the infinitive verb.

In this sentence:

• finite verb: vil
• subject: min datter
• adverb: altid
• infinitives: børste and give

So the order becomes:

• vil min datter altid børste ...

This is very normal Danish word order.

Because they come after the modal verb vil.

After modal verbs in Danish, the next verb is usually a bare infinitive, with no at:

• vil børste
• vil give

So vil min datter altid børste dens pels og give den en godbid is the normal pattern.

Because kaninen is an en-word and is singular.

Predicative adjectives in Danish agree with the noun they describe:

• common gender singular: vågen
• neuter singular: vågent
• plural: vågne

Since kaninen is common gender singular, vågen is the correct form.

With træt, you do not see a difference here, because the common and neuter singular forms are the same: træt.

Because sin/sit/sine refers back to the subject of the same clause.

In the main clause, the subject is min datter. So:

• sin pels would refer to my daughter’s own fur

That is not the meaning.

The fur belongs to kaninen, not to min datter, so Danish uses the non-reflexive possessive:

• dens pels = its fur

So dens is correct here.

Because possessives normally replace the article.

Just like in English you say its fur, not its the fur, Danish says:

• dens pels

not

• dens pelsen

This is completely normal.

Because den is the recipient, and Danish often places a short object pronoun before the thing being given.

So:

• give den en godbid = give it a treat

This is a very common pattern.

You can also say:

• give en godbid til den

But give den en godbid is especially natural when the recipient is a short pronoun like den.

Also, den is used because kanin is an en-word. If it were an et-word, you would expect det instead.

Godbid means treat, tidbit, or goodie.

It is a compound word:

• god = good
• bid = bite / morsel

So a godbid is literally something like a good bite, which matches the idea of a small treat given to an animal.

Yes. Uanset om ... is a very common way to say regardless of whether ...

So:

• Uanset om kaninen er vågen eller træt, vil min datter altid ...

is also good Danish, and it makes the regardless meaning very explicit.

The version with just om ... eller ... is understandable, but uanset om is often clearer if you want to stress that the result is the same in both cases.

Because the sentence begins with a subordinate clause, and that clause is separated from the main clause with a comma:

• Om kaninen er vågen eller træt,
  vil min datter altid børste dens pels og give den en godbid.

This is standard Danish punctuation. The comma helps show where the opening clause ends and the main clause begins.