Learn Icelandic: Borðar strákurinn köku?

Kaka is the dictionary form (the nominative case). In this sentence, the cake is the direct object—it is the thing being eaten. The verb borðar forces its direct object into the accusative case, which changes the ending of the feminine word kaka to köku.

Where is the word for "does" in this question?

Icelandic doesn't use helper words like "do" or "does" to ask questions. Instead, you just swap the order of the subject and the verb. While a statement is Strákurinn borðar (The boy eats), a question simply flips it to Borðar strákurinn? (Eats the boy?).

Does köku mean "cake" or "a cake"?

It can mean both! Icelandic doesn't have a separate word for "a" or "an". Since köku is a bare noun without a "the" ending attached to it, it translates to either "a cake" or just "cake" depending on what sounds best in English.

Why doesn't strákurinn change its ending like köku does?

Strákurinn (the boy) is the subject of the sentence—he is the one doing the eating. In Icelandic, the subject always stays in the default dictionary form, known as the nominative case. Only the object receiving the action (köku) changes to the accusative case here.