If you end up cutting the background, be careful not to cut into the applique piece. (Ask me why I mention this! lol)
You do not have to worry about anything happening to the seams. You cut inside the applique shape and away from the seam allowance. Nothing should come undone because the outline stitching of the applique becomes the boundary seam at that point. If it is a small piece I would not cut anything. Generally, there is no quilting over applique pieces. Quilting is generally done up to the edge of the piece.