It depends on the type of applique you are doing. Sometimes I quilt over my applique, sometimes not. It depends how large it is. If it's a good sized piece, then yes, it needs to be quilted. Smaller sized pieces could go either way. You should definitely quilt right around the edge of the applique. Quilting over it will make it more secure, can add a design to it & can keep it from puffing up too much.
Personally, I don't ever cut away from the back. I know people do, but I'm not really sure why other than that I've been told it makes the quilt lighter.