Learn Spanish (Latin America): ¿Hay un aeropuerto cerca de la ciudad?

We use hay (is there) because we are asking about the existence of an unspecific place (un aeropuerto). If we were asking about a specific place, like 'Where is the airport?' (¿Dónde está el aeropuerto?), we would use está to talk about its exact location.

Why is it cerca de la ciudad and not just cerca la ciudad?

The word cerca (close/near) usually requires the preposition de (of/from) when you mention what the object is near. So, it literally translates to 'close of the city'.

How do we know this is a question? The words are in the same order as a normal sentence.

In Spanish, yes/no questions often have the exact same word order as statements. You tell it is a question by the upside-down question mark (¿) at the beginning, the regular question mark at the end, and a rising tone of voice when speaking.

Why does ciudad use the feminine word la? It doesn't end in 'a'.

While many feminine words end in 'a', words that end in -dad (like ciudad or universidad) are almost always feminine in Spanish. That is why it uses la.