It certainly sounds reasonable that given infinite possibilities there could be some irregular shape out there whose area and perimeter both equal 12 units. It would be interesting to see a mathematical proof one way or the other. It would also be interesting to see if I could follow said proof :-)
No, at least not unless you are going to consider non-planar blankets. If the shape is not specified but we assume it is a planar shape, the minimum perimeter for a region of fixed area is that of a circle of the given area. (This can be proved using a technique called calculus of variations, which is usually only taught in graduate courses, though some top schools will have undergrad courses covering it.) For area 12 square units this gives 2*pi*sqrt(12/pi) which is approximately 12.28.
Only by having the blanket lie on a surface of positive curvature (like the surface of a sphere) could one get the perimeter to be smaller than 2*pi*sqrt(12/pi).