Learn French: La cité antique aurait disparu pendant une tempête. - The ancient city would have disappeared during a storm.

Not exactly. In French news or historical accounts, the past conditional (aurait disparu) is often used to report unconfirmed information. It does not mean it is a failed hypothetical. Instead, it translates better to 'reportedly disappeared' or 'is believed to have disappeared'. The speaker uses this tense to show the fact is not fully proven.

Why does disparaître use avoir (aurait disparu) instead of être?

Even though it describes a change of state or location (which often triggers être, like naître or partir), disparaître always takes the auxiliary avoir in French compound tenses.

Why use antique instead of ancienne for 'ancient'?

Meaning and placement matter here. Une cité ancienne often just means a very old city, and l'ancienne cité means the former city. To specifically mean 'ancient' in the historical and archaeological sense (like Antiquity), you must use antique.

Could I use lors d'une tempête instead of pendant une tempête?

Yes, and it would sound excellent, especially in historical writing. Lors de points to the event as a specific moment in time ('at the time of' the storm), while pendant emphasizes the duration of the storm. Both work perfectly here.