When you cut 4 triangles (quarter square triangles) from a square, the 2 sides that made the "X" when you cut will be bias edges & will be stretchy...but the outside edge will be on the straight of grain, so they'll be more stable on that edge. As long as you're reasonably careful handling the triangles, you should be OK.
I haven't ever used the triangles on a roll, so I can't answer that part of your question. But I'm sure someone here will know! :D
sue