The + on the right is a system-supplied icon, the + on the left is my attempt at an exact replica in Inkscape. I've set the stroke and fill to solid but when I export it as a bitmap I still get parts that are transparent. Are the transparent parts added as part of the export operation, or are they added by Inkscape prior to that? How do I get rid of them for truly solid lines/rectangles?

Probably because your cross doesn't fit on pixel boundaries http://stackoverflow.com/questions/1801 ... l-browsers
try to offset by 0.5px or use https://code.google.com/p/pixelsnap/

Thanks V1nce, in line with your suggestion I've played around with the x-coordinate of my rectangle and have got what I'm after. Cheers.