Learn Icelandic: Hefur þú bókað hótelið?

bóka is a Group 1 weak verb, meaning it ends in -ar in the present tense (bókar). To form the supine (sagnbót) for these verbs, you replace the final -a of the infinitive with -að.

Why are we using both hefur and bókað in this sentence?

This creates the present perfect tense, matching the English Have you booked. You conjugate the helping verb hafa (to have) into hefur to match þú (you), and use the supine form of the main verb, bókað.

Which grammatical case is hótelið in?

It is in the accusative case because it is the direct object receiving the action of being booked. Because hótel is a neuter word, its accusative form looks exactly the same as its nominative (dictionary) form. The -ið ending simply means the.

Why does the verb hefur come before the pronoun þú?

Because this is a yes/no question. Just like in English, we flip the subject and the helping verb. The statement Þú hefur bókað (You have booked) becomes the question Hefur þú bókað? (Have you booked?).