I had the pleasure of seeing three excellent, but different, examples of performing arts over the past week.The first one was among the bloodiest of Shakespeare’s plays. “Richard III,” a dark tale depicting the Machiavellian rise to power and short reign of an English king, is playing through March 3 at the Chicago Shakespeare Theater at Navy Pier. The theater’s new artistic director, Edward Hall, brings to life one of Shakespeare’s most horrific villains — Richard, Duke of Gloucester and brother of King Edward IV …