Yes trim them for sure! I even trim the small ones by taking out that blood line on the tail.
That's what I do too. I keep them alive in the livewell and then before I head in I cut their throats, let them bleed in the livewell for a few minutes, and the motor off. The livewell drains all the blood water out. When I get to the dock it fills up again. When I pull out the livewell drains again leaving it nice and rinsed out. 2 minute drive home and on the cutting board. I trim all the fat and remove the lateral line like Mac Attack said and all is well in Walleye World.