#2
Are you sure you removed the pickup cover that came on your pickup before trying to put the new one on? Because pickups are pretty much 1 size. If you have a strat copy it uses the same size pickups as a strat, so any pickup cover will fit it.
#4
Well some pickup sets have oversized pole-pieces. You could always take and use a dremel to make the holes a bit bigger on your pickup cover if you are careful.

And as far as nut width, it all depends on what taper your neck has. Not even every strat has the same neck width at the nut.