Ég get ekki farið inn án lyklakortsins míns.

After geta (can / be able to), Icelandic usually uses the supine form of the main verb. For fara, that form is farið.

So:

• Ég get farið = I can go
• Ég get ekki farið = I cannot go

You also do not use að here. This is a pattern worth memorizing with geta.

In a normal Icelandic main clause, the finite verb usually comes in second position. Here the finite verb is get, so ekki follows it:

• Ég get ekki farið inn

If you move another element to the front, the verb still stays second:

• Án lyklakortsins míns get ég ekki farið inn.

So the position of ekki is tied to Icelandic word order, especially the verb-second pattern.

Inn is a directional adverb. It means in / inside / into in the sense of movement:

• fara inn = go in / go inside

Inni is used for location, not movement:

• Ég er inni = I am inside

So in this sentence, farið inn is correct because the idea is going in, not being inside.

Án means without, and it takes the genitive case.

That is why the whole phrase after it changes form:

• lyklakort = key card
• án lyklakorts = without a key card
• án lyklakortsins míns = without my key card

This is an important thing to learn in Icelandic: many prepositions require a specific case, and án always takes the genitive.

In Icelandic, the definite article is usually attached to the end of the noun, not written as a separate word.

So:

• lyklakort = key card
• lyklakorts = of a key card
• lyklakortsins = of the key card

So the idea of the is built into lyklakortsins itself.

Because the possessive has to match the noun phrase in gender, number, and case.

Lyklakort is neuter singular, and after án it is in the genitive singular. So minn changes to míns.

Compare:

• mitt lyklakort = my key card
• með lyklakortinu mínu = with my key card
• án lyklakortsins míns = without my key card

So mitt is the wrong form here because the case is different.

In Icelandic, possessives very often come after the noun. This is a common neutral pattern:

• lyklakortið mitt = my key card
• bíllinn minn = my car

When the possessive comes after the noun, the noun is usually definite. That is why you get lyklakortsins míns here.

You can also put the possessive before the noun in some contexts, but that often sounds more emphatic, contrastive, or stylistically different.

Yes. Icelandic forms compound words very freely.

Lyklakort is made from:

• lykill = key
• kort = card

Together, it means key card or access card.

In compounds, the first part often appears in a stem form rather than exactly as it appears in the dictionary, which is why you see lykla- rather than lykill.

Yes, but the word order has to follow Icelandic rules.

For example, you can say:

• Án lyklakortsins míns get ég ekki farið inn.

That is still correct. Notice that:

• the fronted phrase comes first,
• the finite verb get still comes second,
• the subject ég moves after the verb.

That verb-second rule is one of the most important features of Icelandic sentence structure.