Min bror barberer sig hurtigt og tager lidt parfume på, før han går på arbejde.

Because bror is a common-gender noun in Danish.

Danish possessives agree with the gender/number of the noun:

• min = for a singular common-gender noun
• mit = for a singular neuter noun
• mine = for plural nouns

So:

• min bror = my brother
• mit hus = my house
• mine bøger = my books

Sig is the reflexive pronoun for third person.

So barberer sig means shaves himself.

This is very common in Danish with actions people do to themselves:

• vaske sig = wash oneself
• barbere sig = shave oneself
• klæde sig på = get dressed / dress oneself

A useful contrast:

• Han barberer sig = He shaves himself.
• Han barberer ham = He shaves him.
  That means someone else is being shaved.

The full set is:

• mig = myself/me
• dig = yourself/you
• sig = himself/herself/itself/themselves
• os = ourselves/us
• jer = yourselves/you
• sig = themselves

Barberer is the present tense form.

The infinitive is:

• at barbere sig = to shave oneself

In the sentence, Danish uses the present tense because it is describing a habit or routine:

• Min bror barberer sig hurtigt ... = My brother shaves quickly ...

This is very natural in both Danish and English when talking about repeated actions.

Because hurtigt is an adverb describing how he shaves, and in Danish it normally comes after the verb phrase here:

• barberer sig hurtigt

That is the natural order.

So the structure is roughly:

• barberer sig = shaves himself
• hurtigt = quickly

A learner mistake would be something like barberer hurtigt sig, which sounds wrong in normal Danish.

Here tage ... på works like a particle verb, meaning put on or apply.

So:

• tage parfume på = put on perfume / apply perfume

In a main clause, the particle på often comes after the object:

• Han tager parfume på
• Han tager lidt parfume på

This is similar to English:

• put perfume on
• put on perfume

Also note that tage på can mean other things in other contexts, such as gain weight or leave for/go to somewhere, so context matters.

Because parfume here is being treated as an uncountable substance, like water, sugar, or perfume in English.

So Danish uses:

• lidt parfume = a little perfume / some perfume

You do not normally use en when you mean some amount of the substance.

But en parfume is possible if you mean a perfume product / a fragrance:

• Jeg købte en ny parfume = I bought a new perfume.

So:

• lidt parfume = some perfume
• en parfume = a perfume bottle / fragrance product

It is a very common Danish expression meaning go to work.

Literally, English speakers may notice på instead of to, but this is just how Danish says it:

• gå på arbejde = go to work
• være på arbejde = be at work

So you should learn på arbejde as a fixed expression.

Also notice there is no article:

• not på et arbejde
• just på arbejde

That is normal in this idiom.

No. When før means before, it directly introduces a subordinate clause.

So:

• før han går på arbejde = before he goes to work

You do not say før at han går på arbejde in this sentence.

This clause is subordinate, introduced by før. In this example the word order looks simple:

• han går på arbejde

But it is still a subordinate clause. One sign of subordinate word order in Danish is that words like ikke come before the verb:

• ..., før han ikke længere arbejder hjemme
  = ... before he no longer works from home

So før is functioning as a conjunction here.

Because før han går på arbejde is a subordinate clause, and many Danish writers place a comma before that kind of clause.

In modern Danish, there are two accepted comma systems:

• with start comma
• without start comma

So both of these may be seen:

• Min bror barberer sig hurtigt og tager lidt parfume på, før han går på arbejde.
• Min bror barberer sig hurtigt og tager lidt parfume på før han går på arbejde.

Both can be correct depending on the comma system being used. The important thing is to be consistent.

Because the subject is the same for both verbs.

The sentence has one subject:

• Min bror

And two coordinated actions:

• barberer sig hurtigt
• tager lidt parfume på

So Danish does not need to repeat the subject:

• Min bror barberer sig hurtigt og tager lidt parfume på ...

You could repeat the subject in some contexts for emphasis or style:

• Min bror barberer sig hurtigt, og han tager lidt parfume på ...

But that is not necessary here, and the version without the second han is more natural.