Hemlakerfið og hemlaljósið virka.

Because Icelandic usually puts the definite article on the end of the noun instead of using a separate word like English the.

• hemlakerfið = hemlakerfi
  • definite article, the braking system
• hemlaljósið = hemlaljós
  • definite article, the brake light

So Icelandic often says in one word what English says in two.

Because both nouns are definite.

In English, one the can sometimes cover both nouns: the braking system and brake light. In Icelandic, each noun normally shows definiteness itself, so you get:

• hemlakerfið
• hemlaljósið

That is the natural way to say that both specific items are being talked about.

Because the subject is plural in meaning.

Even though hemlakerfið and hemlaljósið are each singular by themselves, they are joined by og (and). Together they form a plural subject, so the verb must be 3rd person plural:

• Hemlakerfið virkar. = The braking system works.
• Hemlakerfið og hemlaljósið virka. = The braking system and the brake light work.

Here virka is the present tense, 3rd person plural of að virka (to work, function).

A useful thing to notice is that this form looks exactly the same as the infinitive:

• infinitive: að virka
• 3rd person plural present: þau/þeir/þær virka

So in this sentence, virka is not an infinitive. It is a finite verb agreeing with the compound subject.

They are in the nominative case because they are the subject of the sentence.

The sentence is saying that these things perform the action/state of virka (work / function), so nominative is what you expect.

They are compound nouns, which are extremely common in Icelandic.

• hemlakerfi = a compound meaning braking system
• hemlaljós = a compound meaning brake light

A very useful rule for Icelandic compounds is this:

• the last part usually tells you the basic meaning
• the last part also usually determines the gender

So:

• kerfi is a neuter noun, so hemlakerfi is neuter
• ljós is a neuter noun, so hemlaljós is neuter

That is why the definite forms are neuter as well: hemlakerfið, hemlaljósið.

Yes, it is the same definite article idea, but the surface form changes depending on the noun.

• ljós → ljósið
• kerfi → kerfið

In ljósið, the ending is clearly -ið.
In kerfið, the noun already ends in -i, so the definite ending appears as -ð attached to that form, giving kerfið.

So both words are definite, but the exact spelling of the ending is not always identical.

This is the normal, neutral word order for a simple statement: subject + verb.

Icelandic main clauses follow a verb-second pattern, which means the finite verb tends to come second in the clause. When the subject is first, the verb naturally comes after it:

• Hemlakerfið og hemlaljósið virka.

If you put another element first, the verb would still stay second:

• Núna virka hemlakerfið og hemlaljósið. = Now the braking system and the brake light work.

So the sentence you were given is the plain, straightforward statement form.

Because Icelandic can refer to the item as a single unit: the brake light.

That may sound slightly different from everyday English, where people often say brake lights in the plural. If Icelandic specifically wanted the plural definite form, it would be:

• hemlaljósin = the brake lights

So hemlaljósið is singular, and that is a real grammatical choice, not just a spelling detail.