You are correct about her being a barnyard mix. It is unlikely that she is carrying the blue egg laying gene because she has a single comb. Also, she is a pullet because of her female-style partridge feathering (males would have black feathers on their chest at this age and have more of a "patchy" look to them) and her small, pink comb.
Lavender does not have anything to do with recessive white. Lavender is a recessive blue dilute gene, sometimes called self-blue. He's got a single comb, so his parents were not pure Ameraucana. Lavender is a project color right now. The stock is still far from 'pure' and they don't breed true yet.