I'd like to get the areas in pixels and the distributions of the different black objects in this image. I assume that with python (openCV or mahotas) it's possible to do it, but I don't know how.
I'd use scikit-image. It looks like a binary image (0s and 1s) to me, so you could use regionprops function to get a list of regionprops objects.
Regionprops ignores 0s (which is black in your case), so you might need to invert the image.