Another person thinking out loud... what about turning it under and sewing it and then using wonder under to apply to the quilt and satin stitching?

I think if using flannel for applique, I would use the method of sewing it to a piece of muslin, right sides together, then making a slit in the muslin and turning it so the flannel is stitched to the muslin all the way around. Press then applique to your square with a zigzag stitch. This should solve the fraying problem there. As for the blocks fraying, I think going with the wider seam and then using the outline stitch for quilting, where you stitch 1/4" from the seamline would secure the seams enough that they wouldn't fray when washed.