Hun svarer høfligt, selv når hun er træt.

Svare is the infinitive, meaning to answer.
Svarer is the present tense, meaning answers or is answering.

Danish often forms the present tense by adding -r:

• at svare = to answer
• hun svarer = she answers

No, it does not need an object here.

In Danish, svare can work like English reply, not just answer something. So Hun svarer høfligt means that she gives her replies in a polite way, without saying exactly what she is answering.

You could add an object or a prepositional phrase in other sentences, for example:

• Hun svarer spørgsmålet. = She answers the question.
• Hun svarer på mailen. = She replies to the email.

But in your sentence, the focus is on how she answers, not what she answers.

Because høfligt is being used as an adverb here.

• høflig = polite as an adjective
• høfligt = politely as an adverb

In Danish, many adverbs are formed from the adjective by adding -t:

• rolig → roligt
• hurtig → hurtigt
• høflig → høfligt

So Hun svarer høfligt literally means She answers politely.

Selv når means even when.

• selv = even
• når = when

Together they introduce a contrast: she answers politely even in a situation where you might expect the opposite.

So the sentence suggests that her politeness continues despite being tired.

Because når is used for general, repeated, or expected situations, while da is used for a specific past event.

Here the meaning is general: whenever she is tired, she still answers politely. That is why når is correct.

Compare:

• Hun svarer høfligt, når hun er træt.
  = She answers politely when she is tired / whenever she is tired

• Hun svarede høfligt, da hun var træt.
  = She answered politely when she was tired
  referring to one particular past situation

Yes, you often could, but the nuance is slightly different.

• selv når hun er træt = even when she is tired
  This sounds more like a repeated pattern or whenever that happens.

• selv om hun er træt = even though she is tired
  This focuses more on the contrast or despite the fact that she is tired.

In many contexts, both are natural. But selv når fits especially well if you mean this is her usual behavior.

Because Danish, like English, usually needs a subject in each clause.

This sentence has two clauses:

• Hun svarer høfligt = main clause
• selv når hun er træt = subordinate clause

The second clause still needs its own subject, so you say hun again.

English does the same thing:

• She answers politely, even when she is tired.

Because træt is an adjective describing hun after the verb er.

This is called a predicate adjective: it comes after a linking verb like er and describes the subject.

• hun er træt = she is tired

With hun, you use the basic singular form træt.

Some related forms are:

• hun er træt = singular
• de er trætte = plural

Also, træt already ends in -t, so Danish does not add another -t.

It depends on which Danish comma style is being used.

Both of these can be correct:

• Hun svarer høfligt, selv når hun er træt.
• Hun svarer høfligt selv når hun er træt.

Many learners will see the comma because it makes the clause boundary clearer. But in modern Danish, leaving out that comma is also common and accepted.

This is a very useful thing to notice, because Danish treats main clauses and subordinate clauses differently.

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

• Hun svarer ikke høfligt.
  = She does not answer politely.

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

• selv når hun ikke er træt
  = even when she is not tired

So compare:

• Hun er ikke træt. = main clause
• når hun ikke er træt = subordinate clause

Yes, but Hun svarer høfligt is the most neutral and natural placement.

You will most often put the adverb after the verb:

• Hun svarer høfligt.

You can also move things around for emphasis, but then the sentence can sound more marked or literary:

• Høfligt svarer hun, selv når hun er træt.

That version is grammatical, but much less neutral. For everyday Danish, the original word order is the best choice.