Learn Icelandic: Hann á kort.

Kort is a neuter noun. A tricky rule about neuter nouns in Icelandic is that their bare plural forms are often exactly the same as their singular forms in both the nominative and accusative cases! Because of this, kort can mean "a card" or "cards". You usually have to rely on context to tell which one is meant.

Is kort in the accusative case in this sentence?

Yes! The verb eiga (to own/have) forces the noun that follows it into the accusative case. For neuter nouns like kort, the accusative form is identical to the nominative (dictionary) form, both in the singular and the plural.

How would I change this to say "He has THE cards"?

To make a plural neuter noun definite (meaning you add "the" to it) in the accusative case, you attach the ending -in. So to say "He has the cards", you would say "Hann á kortin".

Why does this sentence use á for "has" instead of hefur?

In Icelandic, the verb eiga (which becomes á for "he") is used when you physically own or possess a tangible object, like cards, a car, or a book. The verb hafa (which becomes hefur) is mostly used as a helping verb for grammar (like "I have eaten") or for abstract concepts (like "I have an idea").