Den nye kommode er lige så høj som reolen, men ikke lige så bred.

Because kommode is a common-gender noun in Danish, not a neuter noun.

• den is used with common-gender singular nouns
• det is used with neuter singular nouns

So:

• den nye kommode = the new dresser/chest of drawers
• but det nye bord = the new table

This is something you mostly have to learn along with each noun.

In Danish, when a definite noun has an adjective before it, you normally use:

• a separate definite article: den/det/de
• the noun in its basic form

So:

• kommoden = the dresser
• den nye kommode = the new dresser

You do not normally say den nye kommoden.

This is a very common pattern in Danish:

• bilen = the car
• den røde bil = the red car

Because adjectives before a noun usually take -e in a definite phrase.

So:

• en ny kommode = a new dresser
• den nye kommode = the new dresser

That -e ending is very common when the noun phrase is definite and the adjective comes before the noun.

Compare:

• en stor reol = a big bookcase
• den store reol = the big bookcase

Because here they are predicative adjectives: they come after er and describe the subject.

In Danish, predicative adjectives still agree with the noun, but for a common-gender singular noun like kommode, you usually use the basic form:

• kommoden er høj
• kommoden er bred

If the subject were neuter singular, you would usually get -t:

• bordet er højt
• skabet er bredt

If the subject were plural, you would usually get -e:

• kommoderne er høje
• dørene er brede

So høj and bred are correct because kommode is singular and common gender.

It is the Danish pattern for equal comparison: as ... as.

So:

• lige så høj som reolen = as tall as the bookcase
• lige så bred = as wide

The pattern is:

• lige så + adjective + som

Examples:

• Han er lige så gammel som mig. = He is as old as me.
• Bogen er lige så dyr som filmen. = The book is as expensive as the movie.

In your sentence:

• lige så høj som reolen = equal height
• ikke lige så bred = not equal in width

Because Danish uses different words for different kinds of comparison.

Use som with equal comparison:

• lige så høj som = as tall as

Use end with the comparative:

• højere end = taller than

Compare:

• Hun er lige så høj som sin søster. = She is as tall as her sister.
• Hun er højere end sin søster. = She is taller than her sister.

So in your sentence, som is correct because the structure is lige så ... som.

Reol means something like bookcase, shelving unit, or shelf unit, depending on context.

Reolen is the definite singular form: the bookcase / the shelving unit.

For many common-gender nouns, the definite singular is formed with -en:

• en reol = a bookcase
• reolen = the bookcase

Other examples:

• en stol → stolen
• en seng → sengen

So reolen simply means the bookcase/shelving unit.

Because Danish often leaves out words that are already understood from the first part of the sentence.

The full meaning is:

• Den nye kommode er lige så høj som reolen, men den nye kommode er ikke lige så bred som reolen.

But that is repetitive, so Danish shortens it to:

• Den nye kommode er lige så høj som reolen, men ikke lige så bred.

You could also say:

• ..., men ikke lige så bred som reolen.
• ..., men den er ikke lige så bred.

All of these are understandable. The original version is natural because the missing parts are obvious from context.

Because ikke negates the comparison phrase.

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

• Den nye kommode er ikke lige så bred som reolen.

In your sentence, the second part is shortened, and the verb er is omitted because it is understood:

• ..., men ikke lige så bred.

So ikke stays right before the part it is negating: lige så bred.

Yes, that is the natural implication.

If something is not as wide as something else, then it is less wide than that thing. In normal English, we would usually say it is narrower.

So:

• ikke lige så bred som reolen basically means narrower than the bookcase

But the Danish sentence does not use the comparative form directly. It uses a negative equality comparison instead.

Compare:

• ikke lige så bred som reolen = not as wide as the bookcase
• smallere end reolen = narrower than the bookcase

Both express roughly the same idea, but the structure is different.

Yes, in many contexts så ... som is possible, but lige så ... som is very common and often emphasizes that the two things are exactly equally tall, wide, expensive, etc.

So:

• så høj som = as tall as
• lige så høj som = just as tall as / exactly as tall as

In everyday Danish, learners will often hear and read lige så in this kind of sentence, so it is a very useful pattern to learn as a chunk.