You talk about all different kinds of needles, but you don't say what size you're using, and with what thread.
I've solved skipped stitches by going up a needle size.
I've solved shredding thread by using an appropriate needle and needle size for the thread and thread size I'm using. Topstitch needles are good because the eyes are larger and they have a deep groove down the length of the needle, which protects the thread from friction as it moves in and out of the fabric.
The fusible may well be the problem. Maybe you could test this by making a sample sandwich of the same fabrics, without the fusible, and see if you get any skipping.
You mention raising and lowering the presser foot. If you're referring to Dunster's comment about the presser foot height, she's talking about the adjustment that can increase or decrease the amount of pressure the presser foot puts on the fabric. Your owner's manual should have more information about this.