yorunaka : not everyone received - or will receive - a Windlyn beta egg. We got it when Windlyn went to beta. Since you started your account before the switch you received a Windlyn egg. To care for your pet go to: Pet Guide
once you incubate the egg and it hatches you have to keep feeding and playing with it or it dies, so you can either buy and immortality potion from the diamond shop and then it never dies or revive it with a revival potion if it does die. i gave all mine immortality potions ( be careful because the youth potion locks it into a stage and it will never change).