Some times for all your questions. My fabric/cartoon applique is almost always fused. If its small then it will be done on the full piece ie fabric+batting+stabilizer/stiffener. If its large I either do it directly onto the background only with no batting only stabilizer OR I do it off the piece and cut it out and apply it to the backing with either invisible thread of a complimentary thread colour. There is no fixed formula. A large piece wastes the batting if you do it on the quilt sandwich, and I use left over pieces of cotton batting from my quilts constructions.
MaryKatherine