Fiberglass batts: how to cover the exposed paper

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.

Yes, you tape any insulation's lateral seams, especially any fluffy stuff...aluminum HVAC tape lasts the longest. The fluffy stuff really only works best in dead air space. Which also means the un-faced side (where you don't tape anything). If there's a good swap of air or you can feel a breeze when it's windy out, you'll want to seal that out.

1/4" Rigid Foam boards are good for this as they'll seal out most air movement, but still allow humidity to escape into the venting. I think they still have them, the accordion style works very well if you have a number of larger open areas within the truss structure. Otherwise, get the same size as your batts & just put the foam on top.