free motion quilting is done by machine, but its accomplished by you moving the fabric to create a quilting pattern, rather than the machine moving the fabric.
A walking foot is generally used after you've made your quilt "sandwich", that is layered your backing, batting and top. The walking foot helps to move the top layer of fabric at the same time as the lower layer. You would use the walking foot when doing straight line quilting.
You use the darning foot when doing free motion quilting with feed dogs down.
The piecing foot (quarter inch foot) is used when sewing the separate pieces of your block together.
Feed dogs are up when using walking foot.
Feed dogs are down when using darning foot, or free motion foot (although I have read that some people leave the feed dogs up and set the stitch length to zero). I've never tried that.
I use the walking foot to attach binding. I know of no time when you would use a walking foot with the feed dogs down.