I've done this before and the answers to your questions are really more about the specifics of what you want and personal preference. Sewing through the batting is effectively another layer of quilting, and is going to make both the stitching and fabric more secure. But it's harder to do fine detail, so I usually just sew through the backing fabric with a few deeper stitches to tack it in place.
As for washing, holding off will give you a flatter, smoother surface to apply your embroidery too, but it means your stitches need to be able to handle the fabric shrinking in the wash without loosing their place. I'd recommend sticking to a simple running or back stitch if you forgo washing first since those are more secure stitches. If you want to fancier stuff like a chainstitch or stem, then I'd wash the quilt first.