Do periods go inside or outside of parentheses

If you remove parentheses your sentence should be alright. If it lacks logical ending/meaning then You have put the parentheses in wrong place.

jefferson introduced the idea that one does not change a government for "light" causes because he wanted to emphasize the seriousness of colonists' reasons for declaring independence.

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.