Min bror arbejder sjusket, når han har travlt, men min mor er pålidelig og altid hjælpsom.

Because bror and mor are common-gender singular nouns in Danish.

Danish possessives change like this:

• min for common gender singular
• mit for neuter singular
• mine for plural

So:

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

In Danish, just like in English, a possessive normally replaces the article.

So you say:

• min bror = my brother
• min mor = my mother

not something like the my brother.

This is completely normal Danish structure.

Here sjusket is being used adverbially, meaning it describes how he works.

The basic adjective is sjusket, meaning sloppy / careless / untidy. In this kind of sentence, Danish often uses an adjective form where English would clearly use an adverb:

• Han arbejder sjusket = He works sloppily / carelessly

This is common in Danish. Compare:

• Hun synger smukt = She sings beautifully
• De arbejder hurtigt = They work quickly

So even if it looks like an adjective, in this sentence it functions like an adverb in English.

Often, Danish uses the neuter form of an adjective as an adverb-like form, and that form often ends in -t.

For example:

• hurtig → hurtigt
• smuk → smukt
• tydelig → tydeligt

With sjusket, the word already has that shape, so there is no extra change you can see.

So the important idea is: Danish very often uses an adjective form, frequently the -t form, where English would use an adverb ending in -ly.

Here når means when or whenever.

In this sentence, it introduces a situation that is general or repeated:

• når han har travlt = when/whenever he is busy

This suggests that this is something that happens as a pattern, not just one single time.

A useful contrast is:

• når = when/whenever, often for repeated or general situations
• da = when, often for a single event in the past

Because når han har travlt is a subordinate clause.

In Danish:

• main clauses usually follow the verb-second rule
• subordinate clauses usually keep subject + verb order

So:

• Min bror arbejder sjusket = main clause
• når han har travlt = subordinate clause

That is why you get han har, not har han.

A useful comparison:

• Han har travlt = He is busy
• Når han har travlt = When he is busy

At have travlt is a very common Danish expression meaning:

• to be busy
• to be in a hurry
• sometimes to have a lot to do

So:

• han har travlt = he is busy / he is in a rush

Literally, it looks like has busy, but you should learn it as a fixed expression: have travlt.

Examples:

• Jeg har travlt i dag = I’m busy today
• Hun har altid travlt om morgenen = She is always busy in the morning

Because have travlt is a fixed idiomatic expression, and it uses travlt.

The adjective is:

• travl in the common-gender/base form
• travlt in the neuter form

You can see the normal adjective forms in phrases like:

• en travl dag = a busy day
• et travlt liv = a busy life

But in the expression have travlt, Danish uses travlt regardless of who the subject is:

• jeg har travlt
• han har travlt
• vi har travlt

So it is best to memorize have travlt as a whole phrase.

Because the two descriptions are coordinated after the same verb.

So:

• min mor er pålidelig
• (min mor er) altid hjælpsom

The second er is simply omitted because it would be repetitive. English does the same:

• My mother is reliable and always helpful

Both pålidelig and hjælpsom describe min mor.

Because altid is an adverb modifying the adjective phrase hjælpsom.

So:

• altid hjælpsom = always helpful

This is normal Danish word order after er:

• Hun er altid venlig = She is always kind
• Han er ofte træt = He is often tired

In your sentence, altid only goes with the second description:

• pålidelig and altid hjælpsom

That gives the sense:

• reliable, and always helpful

Because they are predicate adjectives describing min mor, and mor is a common-gender singular noun.

In this position, the adjective agrees with the noun:

• min mor er pålidelig
• min mor er hjælpsom

If the subject were neuter, you would normally use -t:

• Barnet er pålideligt
• Barnet er hjælpsomt

So the forms in your sentence match mor, which is common gender singular.

It depends on the comma system being used.

The comma before men is normal and expected, because men links two main parts of the sentence.

The comma before når can vary depending on whether the writer uses start comma or not. So you may see both:

• Min bror arbejder sjusket, når han har travlt, men ...
• Min bror arbejder sjusket når han har travlt, men ...

Both can occur in Danish writing, depending on punctuation style.

Yes. A very natural alternative is:

Når han har travlt, arbejder min bror sjusket, men min mor er pålidelig og altid hjælpsom.

Notice what happens in the main clause after the fronted subordinate clause:

• arbejder min bror

This is normal Danish verb-second word order. When something else comes first, the verb comes before the subject in the main clause.