Ghostrider, thanks for the links - the Tulip site was great. Lots of other good info there.
I agree on your cutting strategy - don't use the lines on the mat, cut using the ruler. I only use the lines as a double-check to make sure I am 'level and plumb' and not cutting parallelograms.
I would be concerned that abrading the surface irregularities would give the mat areas of depressions and it might affect how the rotary cutter passes over them. Passing a cutter across grooves I have noticed makes it stick and parallel/slipping into grooves gets the cutter out of alignment. Maybe I'm just suffering from beginner's fussiness.