Why is Quince the one assigning the roles? Act I, Scene 2
All of these actors/workers have a specific job for which their name is drived. Peter Quince is a carpenter. Although all of these workers are common tradesmen, a carpenter would be socially viewed as slightly above our other workers, thus making him "more important" than those other workers who are more like finishers for a product, i.e. a mender, a joiner, a weaver, etc. Shakespeare was sticking to societal roles when assigning the carpenter as the writer and director of our play within.
Quince is the director of the play, he has determined which person would be best for each part. His choice of Snug to play the lion shows that he is taking into account the strengths of the men - Snug has trouble remembering lines.
At the end of the scene Quince asks them to meet a mile outside of town at the duke's oak that night so they can practice without onlookers. They leave with Quince saying that he will make a list of the props that they will need to present their play.