I use Soft Fuse or Steam-a-seam lite.
As for stitching, I like to use a blanket stitch. As long as I keep the stitch CLOSE to the applique, I tend to not use a stabilizer. However, I have used tear-away stabilizers in the past. You will know fairly quickly if it is necessary as the attached applique will "pull" against the fabric. If you see puckers in the backing fabric, go ahead and use the stabilizer.
I have seen a great way to start and end blanket stitch applique (by Ricky Tims). He starts and ends with a few tiny straight stitches that will be hidden in the straight edge part of the blanket stitch.
When I use zig-zag, I just overlap a few stitches to secure. Never had anything come undone.
For circles, any template will do. Just find one the right size. Go slow when cutting.