There are many different ways to applique. Whether or not you cut away the back can depend greatly on the method you use.
My favorite applique method is machine applique with freezer paper (a la Harriet Hargrave). This method requires cutting away the background behind the applique in order to remove the freezer paper.
Whether or not you quilt over the applique is a matter of personal choice. Typically on an intricate applique quilt you will quilt around the outside of the applique. Sometimes you even want to add trapunto (padding) underneath the applique. With large appliques, you may want to quilt inside the applique to highlight details and to keep it from puffing up. With fusible applique and raw-edge applique, you may want to quilt an all-over design. There are no rules; just how you want the end product to look.