I know of one Alaskan trapper who ties a insulated wire to the jaws and (loosely) together in such a way that it's separated when the trap fires, then checks under-ice conibears with a continuity tester. Search Family Trapper on Trapperman for more information.
I've trapped beaver down to 50 below or so, and a hunk of cardboard (or pile of spruce branches) and some snow over the hole will keep the ice at 6 inches or less for a week in pretty much any conditions. In the spring, I too prefer uncovered holes.
WI must be a cold place to make 30" of ice in a week.