Fordi læben var hævet i går, drak hun kun koldt vand og spiste langsomt.

Because Danish main clauses normally follow the V2 rule: the finite verb must come in the second position.

Here, the whole opening clause Fordi læben var hævet i går counts as the first element. So the main clause has to begin with the finite verb drak, and then the subject hun follows:

Fordi læben var hævet i går, drak hun ...

If the sentence started directly with the subject, then you would get:

Hun drak kun koldt vand og spiste langsomt.

So the inversion is caused by the fronted clause.

Fordi introduces a subordinate clause.

In this sentence, that clause has normal subject + verb order:

læben = subject
var = finite verb
hævet = complement
i går = time expression

So:

Fordi læben var hævet i går

A useful thing to remember is that subordinate clauses in Danish do not use the same V2 pattern as main clauses.

Læben is the definite singular form of læbe:

• en læbe = a lip
• læben = the lip

Danish very often uses the definite form for body parts when the owner is obvious from context. So læben here naturally means her lip, even though hendes is not stated.

This is similar to other common Danish patterns like:

• Hun vaskede hænderne = She washed her hands
• Han brækkede benet = He broke his leg

So læben sounds natural and idiomatic.

Here hævet works as a predicate adjective/state description after var.

So læben var hævet means the lip was swollen or was puffed up.

Historically, forms like hævet can be related to participles, but for a learner the most useful way to read this sentence is:

• var = was
• hævet = swollen

So the sentence is describing a state, not an action being done to the lip.

Because Danish has both irregular/strong and regular/weak past tense patterns.

In this sentence:

• var is the irregular past tense of være = to be
• drak is the strong past tense of drikke = to drink
• spiste is the past tense of spise = to eat

So Danish past tense is not formed in just one single way. Some verbs change vowel, some take endings, and some are highly irregular.

That is why:

• drikke → drak
• spise → spiste
• være → var

Because vand is a neuter noun in Danish: et vand.

When an adjective modifies an indefinite singular neuter noun, it usually takes -t:

• en kold dag
• et koldt glas
• koldt vand

So koldt agrees with vand.

That is why koldt vand is correct, not kold vand.

Because langsomt is being used as an adverb, describing how she ate.

• langsom = slow (adjective)
• langsomt = slowly (adverb)

So:

• en langsom person = a slow person
• hun spiste langsomt = she ate slowly

A very common pattern in Danish is that adverbs are formed with -t from the adjective.

Here kun means only, and its position shows what it focuses on.

In:

drak hun kun koldt vand

the natural reading is that the thing she drank was only cold water.

So kun is limiting the noun phrase that follows it.

Word order with kun can affect emphasis, so learners should pay attention to what comes right after it.

Because the same subject applies to both verbs.

The structure is:

drak hun kun koldt vand og spiste langsomt

The subject hun belongs to both drak and spiste, so Danish does not need to repeat it.

This is the same kind of coordination you see in English:

She drank only cold water and ate slowly.

You could repeat the subject in some contexts, but it would usually sound unnecessary here.

In this sentence, i går most naturally belongs to the fordi-clause:

Fordi læben var hævet i går ...

So it most naturally means that the lip was swollen yesterday.

Because of its placement, it is tied to the reason clause rather than the eating and drinking actions.

If you moved i går, the focus could change. For example:

Fordi læben var hævet, drak hun kun koldt vand og spiste langsomt i går.

That version more naturally connects i går with the main actions.

Yes, the clause can come at the end:

Hun drak kun koldt vand og spiste langsomt, fordi læben var hævet i går.

That version means essentially the same thing, but the emphasis changes a little.
With Fordi ... first, the reason is presented up front.

As for the comma: it marks the boundary between the opening subordinate clause and the main clause. In the given sentence, the comma shows clearly where the fordi-clause ends and the main clause begins.