Bleed everything. If anyone has a trick to bleed crabs I'd love to know it.
Salt water is a coagulant so it actually slows down or stops the bleeding process. Better to lay fish on top of ice if possible or keep in net but out of the water until bled.
I don't like water touching the flesh because the flesh will either absorb fresh water through osmosis or lose water to saltwater through same process so I wouldn't cut the head off but gilling and pulling entrails is a good idea.
Fish tastes best and cooks better after going into and out of rigor mortis so I prefer day old fish over same day.
http://articles.latimes.com/1993-05-13/food/fo-34495_1_fresh-fish