Selv når min søn lader som om, han ikke er træt, putter vi ham efter badet.

Selv når means even when.

It introduces a situation in time: in those situations when this happens.

That is different from selvom, which means although / even though and usually emphasizes contrast more directly.

So:

• Selv når min søn ... = Even when my son ...
• Selvom min søn ... = Even though my son ...

In this sentence, selv når fits well because it describes a recurring bedtime situation.

Når is used for something that happens regularly, typically, or whenever the situation comes up.

Hvis is more like if in a hypothetical sense.

So:

• Når han ikke vil sove ... = When/whenever he doesn’t want to sleep ...
• Hvis han ikke vil sove ... = If he doesn’t want to sleep ... (more hypothetical)

Here, the sentence suggests a familiar pattern, so når is natural.

At lade som om is a fixed expression meaning to pretend or to act as if.

So:

• han lader som om, han ikke er træt = he pretends that he isn’t tired
• more literally: he acts as if he isn’t tired

It is best learned as a whole phrase, because the meaning is idiomatic.

Yes. You can say both:

• lader som om han ikke er træt
• lader som om at han ikke er træt

In modern Danish, the at is often left out, especially in everyday language. The sentence you were given uses the shorter, very natural version.

The part han ikke er træt is a subordinate clause following som om.

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

• lader som om, han ikke er træt

However, Danish comma rules allow two systems. In the system without start comma, this comma can be omitted:

• lader som om han ikke er træt

So the comma after om is normal, but you may also see the sentence written without it.

Because Danish word order changes in subordinate clauses.

In a main clause, the finite verb usually comes before ikke:

• Han er ikke træt.

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

• ... som om han ikke er træt
• ... når han ikke er træt

So han ikke er træt is exactly what you should expect inside this kind of clause.

This is because Danish main clauses follow the V2 rule: the finite verb normally comes in the second position.

The whole opening part:

Selv når min søn lader som om, han ikke er træt

counts as the first element. That means the verb in the main clause must come next:

• ... putter vi ham efter badet

If there were no fronted clause, you would say:

• Vi putter ham efter badet.

So the inversion is caused by putting the Selv når ... clause first.

Because ham is the object form of he.

• han = subject form = he
• ham = object form = him

So:

• Han er træt. = He is tired.
• Vi putter ham. = We put him to bed.

In this sentence, vi is the subject and ham is the object.

Here putter means something like put to bed, tuck in, or settle into bed.

The verb at putte can mean to put/place something somewhere, but with children it very often refers to bedtime.

So putter vi ham efter badet is very natural parental Danish.

Danish often uses the definite form when the reference is already clear from context.

So efter badet means after the bath or after bath time, and in context it is naturally understood as his bath.

English often prefers a possessive in this kind of sentence, such as after his bath, but Danish does not always need one.

Both ideas are possible in Danish, but efter badet sounds very natural here.

Because bad is a neuter noun, and -et is the definite singular ending.

So:

• et bad = a bath
• badet = the bath

Danish usually adds definiteness to the end of the noun instead of using a separate word like the.

Træt is the adjective meaning tired, and here it is used after er.

With a singular common-gender noun like min søn, the basic form is used:

• min søn er træt

Also, træt already ends in -t, so you do not see a different neuter singular form either. The form that changes more visibly is the plural:

• min søn er træt
• mine børn er trætte

So in this sentence, træt is exactly the expected form.