George Bernard Shaw's play, "John Bull's Other Island" is one of several in which Shaw, himself an Irishman who made a literary career in England, first as a critic and then as a playwright, explores Anglo-Irish relationships through the medium of how national character is instantiated in individuals. The types of essay questions an instructor might ask about the play will depend to some extent upon the focus of the class.
One major way in which Shaw differs from most playwrights is that he wrote extensive prefaces to his plays (over 60 pages in the case of "John Bull's Other Island"). His prefaces are as entertaining as his plays, displaying caustic wit and astute analysis of everything from literature to politics. One question an instructor might ask is how reading the preface affected your understanding of the play.
On a similar note, another distinctive feature of Shaw's plays is his extensive stage directions, which read almost as the exposition of a novel. Another question a teacher might ask is how the stage directions, and especially the descriptions of the main characters, add to or detract from the play. How does this affect the difference between seeing a staged performance and reading the play?
On a thematic level, on obvious question would focus on Shaw's understanding of English and Irish national character as illustrated by each of the characters of the play.
Another possible question would focus on postcolonial theory, in particular looking at Doyle and Broadbent through the lenses of hybridity, appropriation, and assimilation.
Another typical question would ask about the gender roles in the play, and particularly whether Nora, like Ireland herself, is treated as a subaltern to be claimed and colonized rather than as an agent herself, in a plot structure that maps colonial onto gender oppression.