What do you do with dead eggs on dragon cave

You don't do anything. They just sit there and there's nothing you can really do about it other than move it to the bottom of your scroll. For the first 24 hours, they count as an egg on your scroll still (so if you have 3 living eggs and 1 egg that just died, you won't be able to take another one for 24 hours). After that, it sits on your scroll for two weeks, and then disappears. This can also be said for dead hatchlings and adults (except the adults are exempt from the rule about taking up a space for 24 hours).