For accuracy, I don't know how you can really use a ruler without using the lines on your mat to line it up all the way across.

I use both mat and ruler for measuring. Sort of depends on what I'm cutting and how it needs to be approached. It's funny how much of this stuff you do without even thinking about it until someone asks a question. There was a thread a couple of days ago about ripping vs. cutting in fabric shops and I was surprised how many people said they preferred to have a fabric purchase cut. I always prefer to have it torn; saves me having to waste fabric establishing a straight edge.

I use Leah Day's way of getting my fabric straight. If you go to Youtube she has a video on how to do it. But basically, you have to hold the selvedges together, but make sure the fold is straight. Move your fabric side-to-side until the fold is even. Then gently lay it down (at this point I press the crease very carefully) then fold in half again, doing the same thing. I always iron the crease because I find I don't get an excess of fabric at the fold. Then I arrange it on the board so that it's even along the bottom and the side, using the lines on the board. I just use 1 ruler to and make sure the fold is even across a mark on the ruler. This works for me.