Learn German: Ist irgendwer dieser Auszeichnung wirklich würdig?

The adjective würdig (worthy) belongs to a specific group of adjectives that require the genitive case. Where English uses the preposition 'of' (worthy of this award), German uses the genitive case directly: dieser Auszeichnung würdig.

Could I use jemand instead of irgendwer?

Yes, Ist jemand dieser Auszeichnung würdig? is perfectly fine. However, using irgendwer emphasizes the 'anyone at all' aspect. It makes the question sound a bit more skeptical, as if the speaker doubts that a single person on earth truly deserves the award.

Why is würdig pushed to the very end of the sentence?

In German, when an adjective governs an object (like the genitive object here), the object is placed before the adjective. Because this is a yes/no question, the conjugated verb Ist takes the first position, and the predicate adjective würdig goes to the very end, framing the rest of the sentence.

Because Auszeichnung is feminine, dieser Auszeichnung could also be dative. How do we know it is genitive?

You are right that the feminine dative and genitive forms look identical. We only know it is genitive here because of the grammatical rule attached to the adjective würdig, which always demands a genitive object.