Backgrounds are generally just static images that are occasionally panned across the stage if there's to be any motion applied to them at all. With Flash especially, animating a BG wastes valuable processing power and diminishes the quality of your overall movie. In that effect, your BG should exist by itself on the lowest layer as a graphic or (unanimated) movie clip. No extra layers or anything.
For the speed lines however, and only because it should be nothing more than a simple looping animation, you could easily nest a FBF movie clip inside one of your backgrounds and use it that way.
If you're wondering how to do it, take a closer look at one of the cartoons you're trying to emulate. On the Japanese ones, they're especially simplified to what looks like maybe a two or three frame loop done on two's.
Hope that helps a little... but to reiterate, BG's should, for the most part, be simple, STATIC images with a lot of time spent on detail.