Should I staple faced insulation batts to the face of the stud or the side?
Attach to the stud face.  The vapor barrier is not as effective if stapled to the stud sides.  And as you say, the fill won't be ideal.  (are your external walls 2x4 or 2x6?)  The paper on the fiberglass is the vapor barrier.  And a barrier is definitely needed where you live.  Optionally, you can use unfaced insulation and a plastic sheet vapor barrier.  I've done lots of sheetrock over insulation and the insulation is not the problem with popped screws.  It can cover and hide forgotten screws/nails/bumps which will cause problems.  One bonus is when there is a couple layers of paper between the sheetrock and studs, it can cut down on noise/vibration conduction through the walls.