Place some newspaper (several layers) over the ice in the cooler and then place your eels in a bucket sitting on top of the ice. They become somewhat dormant and can be easily handled and do not tangle themselves up when placing them on the hook. Once back in the water they become very lively once again.
Bobber or no bobber is a personal preference; weight or no weight as well. Free swimming, without either, seems to work best for me. I do put one on a bobber and one weighted slightly out along with those on the surface. The free swimming eels reach their own determined depth.
We use ice in the summer to help slow them down and to keep them damp. IF they dry up, they'll die in the heat. But recently when eeling the past few weeks, they do just fine in a bucket with a wet towel for 2 days. Use a dry cloth to pick 'em up and they handle easily.
A bucket of sand or a bucket of cat litter works. You dampen your hand and pad it in the sand / cat litter. It gives you an abrasive grip that works well & is renewable. Rags get wet and slimey as the day goes on. Really cheap brown paper towels like you see in public bathrooms work good as well.
Sand works well. When I use to live and surf fish in Nags Head for them, I would take them out of the bucket and drop them in the sand. Then reach down and pick up a handful of sand with the eel. Never had one slide out of my hand when hooking him. The sand always washed off when the eel hit the water.
Come on guys, it's not that bad. Pick up a set of heavy rawhide set of work gloves, always keep a pair in the boat. Slime? No problem. The roughess of the glove(s) will keep the things from slipping out of your hands. I call em "my Johnson gloves"