When exporting to a PDF in Inkscape, blur and transparency are removed, so everything looks quite...ugly. If I were to export it as bitmap, what DPI should I use? I did it at 300 DPI but it looked blurry on the printout. How can I get it to be as clear as it should be?

PDF's don't support Inkscape Blur features and some other SVG features. You can find a number of threads about it on this site.

Exporting at 300 dpi should be more than adequate for printing. But you need to make sure you're printing at the size you expect for the resolution. For example, if you create a 300 dpi image that is 300 x 300 pixels then print it so it's 2 x 2 inches on the page, then the resulting page resolution is only 150 dpi. The dpi in a file isn't a hard-set number - what ever software you are printing with can ignore it or override it. Pixels is the only true way to measure your image, so double check the maths and make sure the pixel size of the image is high enough to print at the size and resolution you want.

i.e. a 2 x 2 inch print at 300 dpi will require the image to be 600 x 600 pixels (2 x 300 = 600).