If I have a long piece of fabric, and I only want a half yard or so off it to cut into smaller pieces, I will rip off that half yard.
Then I press the ruffles down. If I'm cutting small pieces of the 1/2 yard, I just "cut" - it seems to me that tearing first and trimming would waste fabric.
I will also tear the selvages off before sewing backing pieces together. My LAQ suggested making 1/2 inch seams on the backing. So I can live with the torn edges there.