To help with stiffness you can "window Pane" your fusible.
Trace your shape on the fusible and before fusing to the back of your applique fabric, cut out the center of the fusible shape leaving only an approximate 1/4" to 1/2" that will fuse.
If doing a satin stitch I always use a stabilizer of some sort. The kind you use is purely up to you. some people use used dryer sheets, some use a tear away stabilizer, some use a water soluble and some use nothing and heavily starch the background to avoid puckering.
I use a blanket stitch and do not need any stabilizer for that. I have never done it with zigzag.
To start and stop, again it depends what stitch you are using. I remove my block from the sewing machine, flip it over pull both threads to the back and knot them. I have done satin stitch the same way.
Circles are tough to get to look nice and smooth when using a satin stitch. Basically the best way to do circles is to set your machine to the needle down position and stop and rotate your block every few stitches.