I always ask about pets, including if they have any desire for them in the future. Sometimes things change, of course, but if they're planning on getting a cat next month, I'd turn down the job because I don't like cats.
I always ask about any allergies, food or otherwise, and if the children take any medication daily that I would need to administer.
what the family's views are in terms of schedule. When the kids come home from school, how do they unwind? TV? Computer? Books? Outside play? Do they do homework immediately? After an hour? After dinner?
I'm curious -- how is the job full time when all of the kids are in school? Will you be helping out around the house during school hours?