Learn Croatian: Je li direktor bio ovdje tijekom konferencije? - Was the director here during the conference?

It is in the genitive case (konferencije) because the preposition tijekom (during) always requires the genitive. For regular feminine nouns that end in -a, the genitive singular ending is -e.

Where is the auxiliary verb for the past tense in this sentence?

The auxiliary verb je is the very first word. In Croatian yes/no questions, we often use the structure je li + subject + past participle (bio). Here, je acts as both the question starter and the helping verb for the past tense.

Can I use za vrijeme instead of tijekom here?

Yes, you can absolutely say za vrijeme konferencije. Both tijekom and za vrijeme mean during and both take the genitive case. Tijekom is a single word that can sound slightly more professional or formal, but they are used interchangeably.

How would this sentence change if the director was a woman?

The noun direktor would change to its feminine form, direktorica. Because the subject is now feminine, the past tense verb must also agree, so bio would change to bila: Je li direktorica bila ovdje tijekom konferencije?