What part of speech is the underlined word in the sentence? did the people exit the building quickly? quickly is underlined word

quick is an adjective, but quickly, as is underlined, is an adverb

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.