Sometimes you just can't beat ink and paper. To make this work well, you really should draw it by hand then digitize by your method of choice. You are going to spend a lot of time trying to get the fluidity and organic quality of what you see in that example if you try to do it all in the computer.
If you are working from a scanned drawing, you would create a compound path that defines your foreground (the white in your shoelaces example). You can start with a live trace (whatever the Inkscape equivalent is) over a clean scan or simply place the image and start tracing the foreground with the pen.
There are no special production considerations for the areas where the paths overlap. It's just another shape. If an area is blocked off by the overlap shadows then you trace it as a standalone element. If it's a shadow within the bounds of the path, you trace it and then knock it out of the larger shape.
If you use objects of the background color to make the negative space or shadows as you're working just make sure you don't leave it that way in the final. You want to be sure that your art can be placed on various colors and maintain true transparency in those areas. In other words, you should end up with one color on the foreground object, not two that are dependent on the background color as well.