Slightly late response, I'm afraid, but here goes anyway.....
although the boxes have a blank side on top, they have branded images and writing all along the sides of the box. Does this present a problem for mailing the books?
It shouldn't be a problem. I often send out packages (both national and international) using whatever boxes I have to hand.
Also, does anyone know if the UK has such as thing as a 'book rate' - that's how I originally sent all my books over the the US...
There's no equivalent of the U.S. flat-rate envelope for single books, but there is a "Printed papers" rate which is slightly cheaper than standard and can be used for books. There's a low weight limit on it though, about 4 or 5 lb. if I recall correctly, so it wouldn't be any use for a box full of books.
When you get over those sort of weights the cheapest would most likely be ParcelForce International Economy.