Learn Russian: Вы когда-нибудь бывали в Европе или в Азии?

This highlights the general factual meaning of the imperfective aspect. By using бывали (the frequentative form of быть), the question asks about the mere fact of having the experience of visiting these places at any point in your life. While были is also grammatically correct, бывали paired with когда-нибудь strongly emphasizes the 'have you ever had this experience?' nuance.

Why does the word Азия end in -ии here instead of the standard prepositional -е?

Feminine nouns ending in -ия (like Азия, Россия, Италия) are a special group. In the prepositional case, their ending changes to -ии rather than the standard -е. Because Европа ends in a regular -а, it behaves normally and takes the standard -е to become в Европе.

Can I use когда-то instead of когда-нибудь in this question?

No, that would change the meaning and sound unnatural here. когда-то refers to a specific but unnamed point in the past, translating closer to 'formerly' or 'a long time ago'. Because you are asking an open yes/no question about whether an experience has ever happened at any possible time, you must use the indefinite когда-нибудь.

Is it necessary to repeat the preposition в before Азии?