First of all, a hint on transparency. If you export images with transparency for example to PDF (for printing), they are converted to bitmaps inside the PDF or produce unwanted results. Export to non-vector formats should be fine. I suggest to avoid transparency, usually there is a way to mimic transparency, e.g., by just using a light gray for shadows. Also don't use blur effects; similar problems with PDF export.
Now to the checkboard:
It's not possible, one workaround would be to add full page sized box and choose black checker board as fill. Put this box on a separate layer and lock the layer. You might also lighten that box up using one of the filters. Unfortunately this board won't adapt when zooming.
Maybe not the solution, but a related option would be the grid (press #). This is not behind your drawing but in front. It can help you with positioning. Maybe you were looking for this?
