For cutting, I look at what size the background needs, and cut a strip. Then I cut what I need and put the rest of the strip away for the next block that needs that size.
The biggest fabric saver I've found is to use my "EZ angle" and companion angle rulers. There are several different versions of this, I know Bonnie Hunter has her own. But what it does is cut off the dog ear points of right triangles so you don't loose that fabric. So if you need a HST that will be 2.5" UF - you cut a strip at 2.5" wide and then subcut it using that ruler into the triangles. Now, for really small stuff like this, I often cut my strips 1/8" or so wider and then I can sew and trim to exact size.
I know I have partial strips of background of 1.5", 2" and 3.25". I know I'll need them at some point, so I didn't try to use them up. I didn't trim down the 2" strip when I needed 1.5" just to use up what I've already cut. I cut another strip at 1.5" instead of having the waste. It might not seem like much, but 1/4" and 1/2" strip waste can add up when you have a lot of strips!