Learn Hardcore Icelandic: Ég sakna vinkonu minnar þegar hún er í öðru landi. - I miss my friend female when she is in another country.

Because the verb sakna takes the genitive case, not the nominative or accusative.
The basic phrase vinkona mín (my (female) friend) is in the nominative case. After sakna, it has to be in the genitive:

Nominative: vinkona mín – my friend (subject form)
Genitive: vinkonu minnar – my friend (after sakna)

So Ég sakna vinkonu minnar literally means I miss (am missing) of my friend, which is why vinkonu minnar appears in the genitive form.

Why does sakna use the genitive case? Can I use the accusative instead?

No, you cannot use the accusative here. In Icelandic, some verbs always govern the genitive, and sakna is one of them.

Ég sakna vinkonu minnar – correct (genitive after sakna)
Ég sakna vinkona mína – incorrect (that’s nominative/accusative-like ending)

It’s simply a property of the verb: sakna + genitive (of the thing/person you miss). This must be memorised, much like preposition–case combinations.

What is the difference between sakna and missa? Don’t they both mean to miss?

They are quite different:

sakna = to miss someone/something emotionally, to long for
- Ég sakna þín. – I miss you.
missa = to lose (or to miss, as in fail to catch/attend)
- Ég missti símann minn. – I lost my phone.
- Ég missti strætóinn. – I missed the bus (didn’t catch it).

In Ég sakna vinkonu minnar, the feeling is emotional missing, so sakna is the correct verb.

Why is vinkonu minnar in that order? Could I say minnar vinkonu?

The natural order in Icelandic with possessive pronouns like mín is noun + possessive, even when the noun is declined:

vinkona mín – my friend
vinkonu minnar – of my friend

Putting it as minnar vinkonu is not normal Icelandic word order and sounds wrong in this context. So you should keep vinkonu minnar, not minnar vinkonu.

Why is it í öðru landi and not í annað land?

The preposition í takes dative for location (where something is) and accusative for movement (to where something goes).

Location (dative): hún er í öðru landi – she is in another country
Movement (accusative): hún fer í annað land – she goes to another country

In your sentence, she is already in that country (no movement), so you need the dative: í öðru landi.

Why do we say öðru here? What form is that?

Öðru is the dative neuter singular form of the adjective/pronoun annar (other, another).

It has to agree in gender, number, and case with landi:

land is a neuter noun.
Here it is singular.
Because of í (location) it’s in the dative case.

So annar becomes öðru to match landi: í öðru landi = in another country (literally: in other country-dat).

Why is it landi and not land?

Land is a neuter noun, and its dative singular form is landi. After í expressing location, you must use the dative:

Declension of land (singular):

Nominative: land
Accusative: land
Dative: landi
Genitive: lands

So in í öðru landi, landi is simply land in the dative singular.

Why is the pronoun hún used? Is it because vinkona is feminine?

Yes. Vinkona (female friend) is a grammatically feminine noun referring to a female person. The pronoun referring back to her is hún (she).

If you were talking about a male friend, vinur minn (masculine), you would use hann (he):

Ég sakna vinar míns þegar hann er í öðru landi.
I miss my (male) friend when he is in another country.

Can I leave out ég and just say Sakna vinkonu minnar þegar hún er í öðru landi?

Normally, no. Icelandic is not a “null-subject” language like Spanish; you generally need to state the subject pronoun.

So:

Ég sakna vinkonu minnar… – correct and natural.
Sakna vinkonu minnar… – feels incomplete in standard Icelandic; it sounds like a fragment, not a full sentence.

Is þegar hún er í öðru landi the only way to say when she is in another country? What about þegar að?

Þegar hún er í öðru landi is the standard, neutral form. You will hear þegar að in speech (especially informally), but in writing and in careful speech, þegar alone is preferred.

So:

Þegar hún er í öðru landi… – best choice, also what you should learn first.
Þegar að hún er í öðru landi… – colloquial/extra að, often avoided in formal writing.

How would this sentence change if I want to say I miss my friends when they are in another country (plural)?

You need the plural genitive for friends and the plural pronoun þeir (assuming mixed or male group):

Ég sakna vina minna þegar þeir eru í öðru landi.
vina minna = of my friends (plural genitive of vinur minn)

If you specifically mean female friends, you can use:

Ég sakna vinkvenna minna þegar þær eru í öðru landi.
Here vinkvenna is the genitive plural of vinkonur (female friends), and þær is they (feminine).