Hun blev lettet, da lægen sagde, at der ikke var noget alvorligt galt.

Because blev lettet focuses on a change of state: she became relieved at that moment.

• blive = to become
• blev = past tense of blive
• lettet = relieved

So:

• Hun blev lettet = She became relieved / She felt relieved
• Hun var lettet = She was relieved

The second version describes her state; the first highlights the moment the feeling started.

Lettet is an adjective meaning relieved.

In Hun blev lettet, Danish uses blive + adjective very naturally, where English often also uses become/get + adjective:

• Hun blev glad = She became happy
• Han blev træt = He got tired
• Hun blev lettet = She became relieved / felt relieved

So lettet is not a past participle you need to translate literally here; it functions as the adjective relieved.

Here da means when.

In this sentence:

• Hun blev lettet, da lægen sagde ...
• She was relieved when the doctor said ...

For many learners, da is confusing because it can sometimes relate to a cause in other contexts, but here it clearly introduces the time at which she became relieved.

A useful contrast:

• da = when (often about a specific event in the past)
• når = when (more general, repeated, or future situations)

Examples:

• Da jeg kom hjem, spiste vi. = When I got home, we ate.
• Når jeg kommer hjem, spiser vi. = When I come home, we eat / will eat.

So da is the natural choice here because this is one specific past event.

Here at means that.

• lægen sagde, at ... = the doctor said that ...

It introduces a content clause, just like English that.

So:

• Lægen sagde, at der ikke var noget alvorligt galt
• The doctor said that there was nothing seriously wrong

In Danish, at is often used in writing, though in some informal speech it may sometimes be left out, just like English sometimes drops that:

• Han sagde, at han kom senere.
• Han sagde, han kom senere.

Both can occur, but with learners it is safest to include at.

This der is a dummy subject, similar to English there in sentences like:

• There is a problem
• There was nothing wrong

So:

• der ikke var noget alvorligt galt
• there was nothing seriously wrong

It does not mean there as a place here. It is just required by the structure.

Very common Danish patterns are:

• Der er ... = There is/are ...
• Der var ... = There was/were ...

Examples:

• Der er en bog på bordet. = There is a book on the table.
• Der var ingen problemer. = There were no problems.

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

Main clause:

• Hun var ikke glad.
• Here ikke comes after the verb.

Subordinate clause:

• ... at hun ikke var glad
• Here ikke comes before the verb.

So in your sentence:

• at der ikke var noget alvorligt galt

that word order is normal for a subordinate clause after at.

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

Because Danish often uses noget to mean something / anything, and the expression der er noget galt is a fixed, very common way to say something is wrong.

So:

• noget galt = something wrong
• ikke noget galt = nothing wrong / not anything wrong

Then alvorligt adds the meaning seriously:

• noget alvorligt galt = anything seriously wrong / something seriously wrong

In context:

• der ikke var noget alvorligt galt = there was nothing seriously wrong

This is much more idiomatic than trying to build the phrase word-for-word from English.

Because it is the neuter form of the adjective alvorlig.

• alvorlig = serious
• alvorligt = neuter form, and also often used adverbially, like seriously

In this sentence, it is best understood as modifying galt in the sense of seriously wrong:

• noget alvorligt galt = anything seriously wrong

You will often see Danish adjectives in -t when:

1. they agree with a neuter noun, or
2. they are used adverbially.

Examples:

• et alvorligt problem = a serious problem
• Han taler alvorligt. = He speaks seriously

Here the meaning is close to the adverbial use: seriously wrong.

Because galt is the neuter form of gal, and it also appears in the fixed expression noget galt.

• gal = wrong / crazy / mad depending on context
• galt = neuter form

Very common expression:

• Der er noget galt. = Something is wrong.

So in your sentence:

• der ikke var noget alvorligt galt
• there was nothing seriously wrong

This is a standard idiomatic phrase in Danish, and learners should remember noget galt as a chunk.

Because the whole sentence describes a past situation.

• blev = became
• sagde = said
• var = was

The doctor’s statement is reported from the past, so Danish normally keeps that clause in the past too:

• Hun blev lettet, da lægen sagde, at der ikke var noget alvorligt galt.

This matches English quite closely:

• She was relieved when the doctor said that there was nothing seriously wrong.

Danish, like English, often uses backshifting in reported speech when the reporting verb is in the past.

Yes, sometimes, especially in less formal speech or writing.

So both of these are possible:

• Lægen sagde, at der ikke var noget alvorligt galt.
• Lægen sagde, der ikke var noget alvorligt galt.

However, for learners, keeping at is helpful because it makes the structure clearer. In careful writing, many people prefer to include it.

Because Danish punctuation often places commas before subordinate clauses.

Here you have:

• Hun blev lettet, da lægen sagde, at der ikke var noget alvorligt galt.

The commas mark the subordinate clauses introduced by da and at.

Danish comma rules are not exactly the same as English comma rules, so commas can appear where an English speaker might not expect them. If you are learning Danish writing, it is useful to notice that subordinate clauses are often clearly marked with commas.

Yes.

• læge = doctor
• lægen = the doctor

Danish usually adds the definite article to the end of the noun:

• en læge = a doctor
• lægen = the doctor

So da lægen sagde means when the doctor said.

This is a very important feature of Danish nouns:

• en bil = a car
• bilen = the car
• et hus = a house
• huset = the house

It can help to break it into parts:

• Hun blev lettet
  main clause: She became relieved / She felt relieved

• da lægen sagde
  time clause: when the doctor said

• at der ikke var noget alvorligt galt
  content clause after sagde: that there was nothing seriously wrong

So the sentence is:

Main clause + time clause + reported statement

This kind of layering is very common in Danish, and once you identify the clause boundaries, the sentence becomes much easier to understand.