Learn Icelandic: Ferðamaðurinn þarf að staðfesta annaðhvort flugmiðann eða gistinguna. - The tourist needs to confirm either the flight ticket or the accommodation.

No. When used as part of the conjunction annaðhvort... eða (either... or), annaðhvort is a fixed word. It stays exactly the same regardless of the gender, number, or case of the words it connects.

Why are both flugmiðann and gistinguna in the accusative case? Does annaðhvort cause this?

The conjunction annaðhvort... eða does not decide the case. Instead, the verb staðfesta (to confirm) controls the case and requires its object to be in the accusative. Because the ticket and the accommodation are both objects of that verb, they both take the accusative form.

I see the definite articles on flugmiðann and gistinguna. Why do their endings look different?

They have different genders. Flugmiði (flight ticket) is a masculine word, so its accusative definite ending is -nn, giving us flugmiðann. Gisting (accommodation) is a feminine word, which takes the accusative definite ending -una here, giving us gistinguna.

Where exactly do I place annaðhvort in a sentence like this?