I always have my fabric on the straight of grain. That is the way I was taught to sew and I have always followed it since then. Everything fits better, looks better and hangs better.
However, I do not tear my fabric. I pull a thread and then cut along that pull. This gets you a perfect straight of grain. Then, if the sides don't align up correctly, I pull along the bias until the sides do align correctly.
I could not imagine trying to sew on a border that was not on the straight of grain. I would be afraid it would end up all wobbly! Ok, so MAYBE it doesn't make as much of a difference with small pieces, but to me that is kind of like saying an EXACT 1/4 inch seam is not going to make that much of a difference. Maybe that one being off won't, but after 10 or 12 pieces, it will make a difference.