The reason for that box is because some PDF output versions do not support transparencies and transparent blending modes.
A shadow on a color should be in multiply mode, which a standard pdf for print won’t support, so that portion of your pdf is rasterized with the simulated shadow on that pice. (Open your pdf and move the shadow-object and see it for yourself)
1) One way I prepare thoose kind of effects is that I put them all on one layer with the background and I flaten them in a single bitmap prior for exporting.
Warning: I always save it with a new name (MyProject-v35-Output.cdr) so I do not ruin my editable file saving by mistake.
I do that as a rule, but you can confirm your objects are all on CMYK.
2) You probably have a pantone green color, and the overlaped bitmap is always in cmyk. The gap is probably due the conversion from pantone to cmyk.
Another option is to have the black oveprinted in the background color, and setting your pdf output not as cmyk but as original mode. This will output your pantone color below as one plate, and the black, overprinted on it of course.