Ég þarf að læsa hurðinni áður en ég fer, því ég vil ekki skilja íbúðina eftir opna.

Because these two verbs pattern differently.

• þurfa normally takes að + infinitive:
  • Ég þarf að læsa = I need to lock
• vilja normally takes a bare infinitive:
  • ég vil ekki skilja = I do not want to leave

So this sentence is showing two very common Icelandic patterns at once:

• þurfa að gera eitthvað
• vilja gera eitthvað

Because læsa takes the dative case.

In Icelandic, you do not just memorize what a verb means; you often also have to memorize what case it governs.
The pattern is:

• að læsa einhverju = to lock something

So hurð becomes dative singular definite:

• nominative: hurðin = the door
• accusative: hurðina
• dative: hurðinni

That is why the sentence says læsa hurðinni.

No. They are related in meaning, but not the same.

• loka = close
• læsa = lock

So:

• loka hurðinni = close the door
• læsa hurðinni = lock the door

English sometimes blurs these ideas in everyday speech, but Icelandic keeps them separate.

áður en means before.

It introduces a time clause:

• áður en ég fer = before I go / before I leave

The verb fer is present tense of fara, but Icelandic often uses the present tense for future meaning in time clauses, just like English often does:

• before I go
• when he comes
• after we finish

So áður en ég fer is completely normal Icelandic.

It can be understood as either, depending on context.

The verb fara often covers both ideas:

• go
• leave
• depart

In this sentence, because the speaker is talking about locking the door and not leaving the apartment open, fer is naturally understood as leave.

Here því means because.

It introduces a reason:

• ..., því ég vil ekki ... = ..., because I do not want to ...

This is a very common written and spoken connector.
You may also see:

• af því að = because

So the reason part of the sentence could also be expressed with af því að, though því is very natural here.

Because ekki usually comes after the finite verb and before the infinitive or other verb material it negates.

Here the finite verb is vil:

• ég vil ekki skilja = I do not want to leave

This is the normal order.
So Icelandic says:

• vil ekki skilja

not:

• vil skilja ekki

Yes, skilja eftir is a very common verb combination meaning leave behind or simply leave something.

In Icelandic, particles like eftir often get separated from the main verb when there is an object:

• skilja íbúðina eftir = leave the apartment behind / leave the apartment

So the object íbúðina appears between skilja and eftir.

This is very similar to English phrasal verbs:

• turn the light off
• leave the apartment behind

Because íbúðina is the direct object of skilja eftir.

The base noun is:

• íbúð = apartment

The form íbúðina is definite accusative singular:

• íbúðin = the apartment (nominative)
• íbúðina = the apartment (accusative)

Since it is the thing being left, the accusative is used.

No. Here opna is not a verb.

It is an adjective form of opinn = open.

That can be confusing, because Icelandic also has the verb:

• að opna = to open

But in this sentence, opna describes the state of the apartment:

• skilja íbúðina eftir opna = leave the apartment open

So here opna is an adjective, not an infinitive.

Because it agrees with íbúðina.

The noun íbúðina is:

• feminine
• singular
• accusative

So the adjective describing it must match.
That gives opna.

A useful way to think about it is:

• skilja eitthvað eftir opið/opna/opinn
  depending on the gender and number of the thing being described

For example:

• húsið is neuter, so you would get opið
• íbúðina is feminine accusative singular, so you get opna

Because this adjective is being used predicatively/resultatively, not attributively.

Compare:

• opna íbúð = an open apartment
• opnu íbúðina = the open apartment
  
  • here the adjective stands directly before the noun, so you get the weak form

But in your sentence, the adjective is not part of the noun phrase. It describes the state the apartment is left in:

• skilja íbúðina eftir opna

That kind of adjective is usually in the strong form, even when the noun itself is definite.

Because it belongs to the whole expression skilja íbúðina eftir opna.

The order is:

• verb: skilja
• object: íbúðina
• particle: eftir
• result/state adjective: opna

So the sentence builds up like this:

• skilja íbúðina eftir = leave the apartment
• skilja íbúðina eftir opna = leave the apartment open

Putting opna at the end is the natural Icelandic order here.