Ultimately, you'd have to look up the history of the suit, but a double breasted lounge suit is a step-up in formality from the single breasted version. The peak lapel is a step up in formality from the notch lapel. It's a sartorial mismatch to have a notch lapel with a double breast. You do see it occasionally, but it's more of a fashion move than anything else. The shawl collar is is found on DB dinner jackets, mostly.
Lastly, DB suits are designed to present a specific visual aesthetic, which is the broadening and lengthening of the torso. The different button stances allow for different builds of torso. The peak lapel is a natural fit for that purpose.
I don't think this is the conventional wisdom on DB v SB formality. Will from A Suitable Wardrobe posted about this very subject here: http://asuitablewardrobe.dynend.com/2010/10/formality-is-principally-in-details.html