I don't think it needs borders. The black/white strip binding right against what you have now would look fabulous! I can't see it with anything else for borders - any colour pulled from the quilt feels like it would overwhelm what you have - so if you really want borders, I'd just go with the white background.
For backing, I'd pick a cute kids print that picks up some of the bright colours in the front. But really, almost anything would work. I guess since the binding is black & white, it would be nice if there was some of that in the backing. The fabric in purple at the bottom of the second photo, for example, would be cute on the back. But you don't have to match any of the circles.