I'm glad you're liking the game. And, yes, by "acquire" a peasant we mean either produce one from your Town Hall or hire one from one of the NPC towns.
Quests can't be restarted or skipped (although that is an option we might consider adding in sometime in the future), but the expansion quest should work, no matter when you start it (provided you're not at the house limit).
It's possible there is a bug that is preventing the quest from completing. The bug might correct itself the next time you play, however, if it doesn't, can you please exit the game then reply here with your log1.txt file attached? You'll find the file in the folder in which the game is installed.