If you have stained built-in bookcases, I would stain the trim and crown molding to tie the room together.

Will it be odd having only one room, in this case the family room, with stained molding in sight of the hallway, kitchen and study where the moldings and baseboards are all painted?

Depends. Look at my Peacock Chinoiserie. Obviously the cabinetry is VERY DIFFERENT from adjacent crown, yet it bridges elements from 5 other rooms! Do you want your area to stand out too? Consider keeping the crown and baseboards the same as the other others to tie rooms together. Consider the stain or paint to be a color that compliments another detail from another room. You can do this!