Whenever you catch an egg in the biomes, (it doesn't apply to self bred dragons or to dragons caught in the Abandoned Pile (AP) or traded eggs), you must keep the egg in your posession for 5 hours before you can abandon or trade/gift it away.. Although you can waste a kill action and kill it immediately, but it will count towards egg limits for 24 hours..
Woohoo, I have some rares and uncommons, unfortunately I couldn't figure the dc forums to save my life.
If someone has a trade set up, you'd make an offer by clicking a link they show. The person who started it then can accept, and their egg is yours.