How about a marathon endurance trophy?
One gets it when playing all the time, so making a move in every game and not taking a break bigger than, say 10 minutes (to go to the toilet).
No, that will have a negative effect on the health of people on lichess.
(Nice captcha!
http://en.lichess.org/iGIkKuCc/black Not every day you can checkmate with a king move.)
I don't think that can be called a reason. Everybody should be able to judge himself if he can do that physically.
I do not suggest rewarding people for doing something that physically harms them.
Why should there be big physical harm? I mean, you don't just die just because you don't sleep for 24h.
In a 48h marathon you might have a point but not in the 24h one.
Besides: People also get rewarded for professional or whatsoever even if it's clear that this causes BIG damage to your body. So I generally don't see the point here.
EDIT: *professional boxing
Where is the edit button?