If you were to 'color' these like a four color map problem, are they all trivially colorable with just two colors? It looks that way from the pictures, but I lack the math intuition to prove it.
Yes, the intersection of a number of curves defining regions is color-able with two colors: any point covered by an odd number of regions is black and any point covered by an even number of regions is white (i.e. a checkerboard pattern). Works as long as the boundary intersections are transverse. The proof is simply that if you cross a boundary, you must be decreasing or increasing the region count by one, changing the parity.
Each color is touched by at most one color on top and one color on bottom. Looking at the original 7-Venn we get Red->Orange->Yellow->Green->Blue->Purple. Change the odd colors to white and the even colors to black and you're done.