Eyrnalokkarnir passa vel við hálsmenið sem amma gaf henni.

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

• eyrnalokkur = earring (singular)
• eyrnalokkar = earrings (plural)
• eyrnalokkarnir = the earrings

The ending -nir here is the definite article for a masculine plural nominative noun.

For the same reason: the definite article is attached to the noun.

• hálsmen = necklace
• hálsmenið = the necklace

The ending -ið is the definite article for a neuter singular noun.
So Icelandic says hálsmenið where English says the necklace.

Passa við means to match, to go with, or to suit.
Adding vel means well.

So:

• passa við = match / go with
• passa vel við = go well with / match well

In this sentence, Eyrnalokkarnir passa vel við hálsmenið means that the earrings and the necklace look good together.

Yes. In this meaning, við takes the accusative.

So after passa vel við, the thing something matches with appears in the accusative case:

• passa vel við hálsmenið = go well with the necklace

Here, hálsmenið looks the same in nominative and accusative because it is a neuter singular noun, so you do not see a change in form.

Sem introduces a relative clause, like English that, which, or sometimes who.

So:

• hálsmenið sem amma gaf henni
  = the necklace that grandma gave her

A useful thing for learners: sem does not change form for gender, number, or case. It stays sem.

Because the relative word sem already stands for it.

In English, we say:

• the necklace that grandma gave her

Here, that refers to the necklace, which is the thing being given.

The same thing is happening in Icelandic:

• hálsmenið sem amma gaf henni

The missing direct object after gaf is understood to be hálsmenið.
So the structure is essentially:

• Grandma gave her [the necklace]

but the noun is not repeated because sem links back to it.

Because this is a subordinate clause introduced by sem.

In a main clause, Icelandic often follows the verb-second pattern. But in subordinate clauses, that pattern does not work the same way, and you commonly get a more straightforward order like:

• amma gaf henni
  = grandma gave her

So the relative clause order here is normal and natural.

Because gefa (to give) usually takes:

• a direct object = the thing given
• an indirect object in the dative = the person receiving it

So in:

• amma gaf henni

henni is the dative form meaning to her / her.

Compare the pronoun forms:

• hún = she
• hana = her (accusative)
• henni = her / to her (dative)

With gefa, the recipient is dative, so henni is exactly what we expect.

Because amma is the subject of the relative clause, and subjects are normally in the nominative.

In sem amma gaf henni:

• amma = the person doing the giving, so it is the subject
• therefore it appears in the nominative form amma

The form ömmu is a different case form of the same noun and would be used in other grammatical roles.

Icelandic does not use a separate word like the before nouns in the same way English does, and names or family-role words used almost like names often appear without any article.

So amma here is simply grandma, referring to a known person from context.
This is very natural in Icelandic.

The nouns are:

• eyrnalokkur = masculine
• hálsmen = neuter
• amma = feminine

This matters because gender affects things like:

• the form of the definite article
• adjective endings
• some pronouns and case forms

That is why you get different definite endings:

• eyrnalokkarnir with masculine plural -nir
• hálsmenið with neuter singular -ið

Here it is genuinely plural: the earrings.

That is natural, because earrings are usually talked about as two separate items worn as a pair. Icelandic, like English, commonly uses the plural for that.

The singular also exists:

• eyrnalokkur = an earring

But in this sentence, the plural makes sense because the sentence is talking about the earrings as a set matching the necklace.