Hvis pæren går i stykker, virker lampen ikke.

In Danish, the definite article is usually added to the end of the noun.

• en pære = a bulb

• pæren = the bulb

• en lampe = a lamp
• lampen = the lamp

So the -en ending is doing the job that the does in English.

Hvis means if and introduces a condition.

So:

• Hvis pæren går i stykker = If the bulb breaks / if the bulb stops working

This creates a subordinate clause, which affects the word order in the rest of the sentence.

Gå i stykker is a very common Danish expression meaning to break, to get broken, or to stop functioning.

Literally, it looks like go into pieces, but you should learn it as a fixed expression.

Examples:

• Min telefon gik i stykker. = My phone broke.
• Vasen gik i stykker. = The vase broke.

In your sentence, pæren går i stykker means the bulb breaks or burns out.

Because the sentence starts with a subordinate clause:

• Hvis pæren går i stykker, ...

After that clause, the main clause follows normal Danish verb-second word order. That means the finite verb comes before the subject when something else has been placed first.

So:

• Hvis pæren går i stykker, virker lampen ikke.

The main clause by itself would normally be:

• Lampen virker ikke. = The lamp doesn’t work.

But after the hvis-clause comes first, Danish flips the order:

• ..., virker lampen ikke.

This inversion is very important in Danish.

In a main clause, ikke usually comes after the subject and after the finite verb structure has been set up.

Here the main clause is:

• virker lampen ikke

Word by word:

• virker = finite verb
• lampen = subject
• ikke = not

So the negation comes after the subject.

Compare:

• Lampen virker ikke.
• I dag virker lampen ikke.
• Hvis pæren går i stykker, virker lampen ikke.

In all of these, ikke comes late in the clause.

No. Virke can mean different things depending on context.

Common meanings:

• to work / function
• to seem / appear

In this sentence, it clearly means function:

• Lampen virker ikke. = The lamp doesn’t work.

But in another sentence:

• Han virker træt. = He seems tired.

So context tells you which meaning is intended.

Yes, and the meaning is close, but the nuance is slightly different.

• Hvis pæren går i stykker = If the bulb breaks
  This focuses on the event of it breaking.

• Hvis pæren er i stykker = If the bulb is broken
  This focuses on the state of being broken.

Both are natural, but går i stykker emphasizes the change, while er i stykker describes the result.

Because Hvis pæren går i stykker is a subordinate clause, and in Danish it is standard to separate it from the main clause with a comma.

So the structure is:

• subordinate clause: Hvis pæren går i stykker
• main clause: virker lampen ikke

The comma helps show that boundary.

The letter æ is a front vowel that does not exist in exactly the same way in standard English, but it is somewhat similar to the vowel in cat for many learners.

Very roughly:

• pære sounds something like PEH-reh
• pæren sounds roughly like PEH-ren

A few points:

• æ is more open than the vowel in say
• the r affects the vowel quality in many Danish accents
• the final -en is usually unstressed

The exact pronunciation depends on accent, but recognizing æ as its own Danish vowel is important.

No. Pære can also mean pear, just like English sometimes uses the same word shape for unrelated meanings.

Context tells you which one is meant.

• en pære = a bulb
• en pære = a pear

In your sentence, because of lampen, it can only mean the bulb.