Hún gleymdi lyklakortinu heima, svo yfirmaðurinn opnaði hurðina fyrir hana.

Because gleyma takes a dative object in Icelandic.

• lyklakortið = the key card in nominative/accusative
• lyklakortinu = the key card in dative

So after gleymdi you need the dative form:

• Hún gleymdi lyklakortinu = She forgot the key card

This is something English speakers often have to memorize verb by verb.

Yes, normally gleyma is used with a dative object.

A useful pattern to learn is:

• að gleyma einhverju = to forget something

Examples:

• Ég gleymdi símanum. = I forgot the phone.
• Við gleymdum nafninu. = We forgot the name.

So in your sentence, lyklakortinu is dative because of gleyma.

Heima means at home, so it describes location.

• heima = at home
• heim = home, homeward, to home

In this sentence, she forgot the card at home, so heima is correct.

Compare:

• Hún er heima. = She is at home.
• Hún fór heim. = She went home.

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

• yfirmaður = supervisor / boss
• yfirmaðurinn = the supervisor / the boss

The same thing happens elsewhere in the sentence:

• lyklakortinu = the key card
• hurðina = the door

So Icelandic often expresses the as a suffix, not as a separate word.

There is, but very often it appears as an ending attached to the noun rather than as a separate standalone word.

In this sentence:

• yfirmaðurinn = the boss
• hurðina = the door
• lyklakortinu = the key card

English speakers often expect a separate word before the noun, but Icelandic usually builds definiteness into the noun itself.

Because hurð is a feminine noun, and here it is the direct object of opnaði.

The verb opna takes an accusative object:

• að opna hurð = to open a door

With the definite article, the accusative singular becomes:

• hurðina = the door

So:

• yfirmaðurinn opnaði hurðina = the boss opened the door

Because in this sentence fyrir means for / on behalf of, and with that meaning it takes the accusative.

So:

• hana = her (accusative)
• henni = her (dative)

That is why the sentence says:

• fyrir hana = for her

This is a good reminder that Icelandic prepositions often control a specific case, and sometimes the case changes with the meaning.

Yes. They both refer to the same woman, but they are in different cases.

• hún = she (nominative, used for the subject)
• hana = her (accusative)

So:

• Hún gleymdi... = She forgot...
• ...fyrir hana = ...for her

English mostly keeps the same noun unchanged, but Icelandic pronouns change form depending on their grammatical role.

Here svo means so, introducing the result of the first clause.

• Hún gleymdi lyklakortinu heima = She forgot the key card at home
• svo yfirmaðurinn opnaði hurðina fyrir hana = so the boss opened the door for her

Depending on context, svo can also mean then, but here so is the natural meaning.

Because svo is functioning here as a conjunction linking two clauses, like English so.

That means the second clause keeps normal clause structure:

• svo yfirmaðurinn opnaði hurðina fyrir hana

If svo were being used more like an adverb meaning then, you could also see:

• Svo opnaði yfirmaðurinn hurðina fyrir hana.

That version feels more like Then the boss opened the door for her, while your sentence is more clearly She forgot the card, so the boss opened the door for her.

Lyklakort is a compound noun meaning key card or access card.

It is made from:

• lykla- = keys / of keys
• kort = card

Icelandic makes a lot of compound nouns this way. So instead of using several separate words, it often combines them into one.

Because the sentence contains two separate clauses:

• Hún gleymdi lyklakortinu heima
• yfirmaðurinn opnaði hurðina fyrir hana

The comma helps separate them clearly when they are joined by svo. That punctuation is very natural in Icelandic here.

Yes. Yfirmaður can mean boss, supervisor, or sometimes manager/superior, depending on context.

So the sentence could be understood as:

• the boss opened the door for her
• the supervisor opened the door for her

The exact English choice depends on the situation, but the Icelandic word itself is perfectly natural here.