There are several different types of machine applique. It sounds as if you are doing fusible with a blind hem stitch, correct?
It is best to cut out the middle of the fusible for less stiffness in the applique. However, when you do this, it is not necessary to cut out the background fabric afterwards. That is usually done with freezer paper machine applique in order to remove the freezer paper.
Fusible applique with a blind hem machine stitch is usually done with two strands of black thread to mimic old-fashioned hand sewing stitch. This stitching is purely decorative, since it won't necessarily keep the cut fused edges from raveling in the wash.
I agree that using YLI nylon monofilament or similar thread would keep the stitches from showing; however, I'm not sure what the purpose would be. Monofilament is used for invisible machine applique done with freezer paper, which creates a turned edge similar to hand applique. In that case, the stitching is necessary to fix the applique to the background. Blind-hem stitching around a fusible is primarily decorative, as I mentioned above, so ordinarily you would want the stitches to show.
If you want to completely cover the fused edges to prevent raveling, you can use a satin stitch. When using a satin stitch, however, you have to either heavily starch the background fabric or use a stabilizer under the background fabric to prevent "tunneling".
For a fusible whose edges are not going to be satin stitched, in my experience Steam-a-Seam is the best fusible because it is less likely to ravel at the edge.
I don't use fusibles very often. I prefer invisible machine applique using freezer paper.
Usually it is a good idea to back a white piece with another layer of fabric (or even a layer of thin batting) so the background color doesn't show through. Whites in particular seem to shadow through, especially when the background is dark.
Edit: Oh, come to think about it, cutting out the background fabric underneath the white would probably have a similar result.