Lykillinn er fastur, svo ég get ekki opnað hurðina.

Lykillinn is lykill (key) with the definite article attached: -inn = the.

• lykill = a key / key
• lykillinn = the key

Icelandic usually puts the as a suffix on the noun (instead of a separate word like English).

It’s nominative singular because it’s the subject of the sentence (the thing that is something):
Lykillinn er fastur = The key is stuck.
With the verb vera (to be), the subject stays in the nominative.

Because adjectives agree with the noun they describe in gender, number, and case.
lykill is masculine, singular, nominative, so the adjective takes the masculine nominative singular form:

• masculine nom. sg.: fastur
  (You’ll see different forms with other genders/cases, e.g. föst / fast / föstu, depending on context.)

In this context it means stuck (as in jammed and not turning/coming loose).
While fastur can sometimes translate as fixed/firm/secure, with things like keys, lids, screws, etc., it commonly means stuck/fast.

Because svo here works like so / therefore, linking two clauses:
Lykillinn er fastur, svo ... = The key is stuck, so ...
In Icelandic, it’s normal to put a comma between the first clause and the result clause introduced by svo.

Yes. When svo means so/therefore and introduces a main clause, it typically triggers verb-second (V2) word order: the finite verb comes right after the first element.
So after svo, the first element is ég, and the finite verb get comes next:
svo ég get ... (not svo ég ekki get ...)

ég get ekki opnað means I can’t open (inability). It uses:

• geta = can / be able to (finite verb: get)
• a second verb in the infinitive: opna or opnað (see next question)

ég opna ekki would mean I don’t open / I’m not opening (simple negation of the action), which is a different idea from not being able to.

Both can be heard, but geta commonly takes either:

• the infinitive: geta ekki opna
• or a form often called the supine/past participle in this construction: geta ekki opnað

In everyday Icelandic, get ekki opnað is very common and natural for can’t manage to open / can’t get open.

In this kind of sentence, ekki normally comes after the finite verb (get) and before the non-finite verb/object:
ég get ekki opnað hurðina.

You generally don’t put ekki before get in standard word order.

Because hurðina is accusative singular definite: the door as the direct object of opna (to open).

• hurð = a door (indefinite base form)
• hurðin = the door (nominative, typically subject)
• hurðina = the door (accusative, typically object)

Since you’re opening it, it’s the object → accusative.

The sentence doesn’t explicitly say in the lock, but it strongly suggests a common scenario: the key is stuck (in a lock) or jammed so you can’t use it to open the door. If you needed to be explicit, you could add a location phrase (e.g., “in the lock”), but the given sentence is natural as-is.

A rough guide (not perfect IPA, but helpful):

• Lykillinn ≈ LIH-kih-tlin (with a “tl” sound)
• er ≈ ehr
• fastur ≈ FAH-stur
• svo ≈ svoh
• ég ≈ yeh-gh (a soft “gh” at the end)
• get ≈ geht
• ekki ≈ EH-kih
• opnað ≈ OP-nath (ð is often a soft “th” sound)
• hurðina ≈ HUR-thih-na (ð ~ “th”)