Lægen siger, at symptomerne ikke er alvorlige, men at jeg skal hvile mig.

Because Danish usually puts the definite article at the end of the noun.

• en læge = a doctor
• lægen = the doctor

So -en here is the definite ending for a common gender noun.

Symptomerne means the symptoms, and it is built in steps:

• et symptom = a symptom
• symptomer = symptoms
• symptomerne = the symptoms

So the word contains:

• the noun stem symptom-
• the plural ending -er
• the definite plural ending -ne

This is very common in Danish.

The first at introduces the clause after siger:

• Lægen siger, at ... = The doctor says that ...

The second at is repeated after men to show that the second part is also something the doctor says:

• ..., men at jeg skal hvile mig = ..., but that I need to rest

This makes the structure clear and balanced:

• The doctor says [that X], but [that Y].

If you leave out the second at, the sentence structure changes, and it can sound more like the second clause is the speaker's own statement rather than part of what the doctor says.

Because this is a subordinate clause, and in Danish, sentence adverbs like ikke usually come before the finite verb in subordinate clauses.

So you get:

• at symptomerne ikke er alvorlige

But in a main clause, the order is different:

• Symptomerne er ikke alvorlige.

This is one of the most important word-order differences in Danish.

A useful contrast:

• Main clause: subject + verb + ikke
• Subordinate clause: subject + ikke
  • verb

Because the adjective agrees with the noun it describes.

Here, symptomerne is plural, so the adjective takes the -e form:

• et symptom er alvorligt = a symptom is serious
• symptomer er alvorlige = symptoms are serious
• symptomerne er alvorlige = the symptoms are serious

So alvorlige is the correct plural form here.

Here skal means something like must, have to, or need to.

In this sentence, it sounds like a doctor's instruction or recommendation:

• jeg skal hvile mig = I need to rest / I have to rest

A few comparisons:

• skal = stronger obligation or instruction
• bør = should / ought to (softer)
• må = may / am allowed to / might, depending on context

So skal is very natural when reporting what a doctor says you need to do.

Because Danish often uses hvile sig as a reflexive expression meaning to rest.

So:

• jeg hviler mig = I am resting / I rest
• jeg skal hvile mig = I need to rest

This is idiomatic Danish. English usually just says rest, but Danish often includes the reflexive pronoun.

Because the subject is jeg (I), so the reflexive/object form is mig.

Danish uses:

• jeg ... mig
• du ... dig
• han/hun/man ... sig

So:

• jeg skal hvile mig = correct
• han skal hvile sig = correct

Sig is only used for third person subjects.

Often, yes: in a subordinate clause introduced by at, the normal order is:

subject + sentence adverb + finite verb + rest

For example:

• at jeg ikke kommer
• at hun aldrig spiser morgenmad
• at symptomerne ikke er alvorlige

But if there is no adverb like ikke, the clause may look more like English:

• at jeg skal hvile mig

So the key thing to remember is that subordinate clauses often place ikke, aldrig, ofte, etc. before the finite verb.

Because at symptomerne ikke er alvorlige is a subordinate clause, and Danish often places a comma before such clauses.

So:

• Lægen siger, at symptomerne ikke er alvorlige ...

This is normal written Danish.

A small extra note: modern Danish allows two comma systems, so you may sometimes see the comma omitted before at in some writing styles. But the version with the comma is very common and especially helpful for learners.

Yes, it is completely natural.

It may feel a little unusual to an English speaker because English sometimes drops the second that:

• The doctor says that the symptoms are not serious, but that I should rest
• or simply ..., but I should rest

Danish often keeps the second at when two subordinate clauses are joined, especially when both depend on the same verb, here siger. It makes the sentence clearer and more symmetrical.

It breaks down like this:

• Lægen siger = main clause
• at symptomerne ikke er alvorlige = first subordinate clause
• men at jeg skal hvile mig = second subordinate clause

So the pattern is:

Main clause + subordinate clause + conjunction + subordinate clause

Or more simply:

The doctor says [that X], but [that Y].

Seeing the sentence in parts can make it much easier to understand and build similar sentences yourself.