They will eventually drown themselves in a skim of water; its hard to believe but they manage.
Putting them on the hook. First - if you subscribe to the Mike F. approach, icing them down calms them long enough to get them hooked. Me, its simple; I try and put the bait rag over their head and perhaps 1/2 of their body. It calms them. Then I grab the body somewhere in the upper third hard thru the rag. This causes them to squirm backwards When the head gets to my grip I tighten even more; roll the bait rag off their head and hook quickly into the lower jaw and out an eye socket. If I miss I put them down a minute, wait and start over again.
Once I have them hooked I get them in water within a couple seconds, not
by casting, but by letting them swim in shallows by just dipping the rod tip down to the water. Let them swim a minute, then cast anywhere once or twice. Now they are calm enough that you can position yourself and start aiming your casts as you desire.
Never ever hook up an eel on the rocks if you can find a sandy spot or grassy bank nearby. Dropping them on a bank or beach is recoverable, dropping them in rocks is a lost eel. I've only fished from mud bank areas of marshes, but I could see how dropping one in grass is a guranteed lose.
I hook mine at my pond on the bea c h by the middle of the jetty, half way from the pond entrance and halfway from the ocean entrance. Once I've made two or three casts I work my way to either end about 10 yards per cast, when the eel comes out of the water,I advance 8 or 10 rocks, and get them in the water before they start to ball up.
I rarely drop eels anymore; my problem which I have mostly cured, was taking too much time on land w/ a wiggly eel and ending up with a ball of slime, eel, and line. You think wind knots are bad, eel knots are worse...