Thanks everyone.
Re: #1, We did require players to have both the egg AND the quest in order to continue hatching the dragon.
We made the quests and adventure expire at the same time to try and avoid confusion, however this seems to have caused extra confusion, as anyone who got an egg after everything expired cannot use it. I'll see what I can do about this, but I can't make any promises on a public forum without discussing things with the team.
Re: #2: this is not good! So basically, ANYONE who had a quest expire was able to move on to the next step, despite not having the egg? I can see why this would cause a lot of confusion. Thank you. I'll investigate and see what can be done.