Why must the region ruled
by Hades had been a very
gloomy place?

because Hades was the god of the underworld

Explanation:

He was the god of death so he lived in the underworld and ruled it. If I were dead I would probably be gloomy too.

the sentence that avoid faulty coordination is:

culture, which is well represented in philadelphia, plays a major role including theatre, music, and the visual arts.  

faulty coordination occurs when the writer gives equal importance two different ideas (clauses) which in fact are not equivalent. in order to correct a misplaced clause or idea, the writer can place less important or emphatic ideas in a relative clause as in the previous sentence.