Especially for children's quilts, some people apply the binding to the back of the quilt, and then top stitch it down on the front.
Sometimes it helps to make a sandwich (a mini front, batting, and back) and experiment (with scrap materials) to get the width you want.
I highly recommend using the two layer binding. If/When the top layer wears out, there's still a layer of fabric to protect the edges.
I also recommend using a bias cut binding on a child's quilt - I"ve heard that it wears a bit better because more than one thread is exposed at the edge.