What you say about making the neck hole bigger makes perfect sense. You can make it as large as you need to. I would pick up all the stitches around the neck to start the cowl, but then gradually reduce the stitches after about an inch to the neck doesn't turn out too floppy.
I always bind off with a needle one size larger, but you can use any size needle that makes it look the way you want. Bound-off edges are usually woven together or picked up, so as long as they're stretchy enough, do whatever works.