Ross, most dogs are coprophagous, when it comes to cat leavings. I've read that it's because cats leave a lot of protein in their droppings.
Although our dogs know we don't want them to do it, they like to have lives of their own, to which we're not entirely privy. I believe that as far as this abberation, they think that what we don't know won't hurt us. They don't realize the flecks of litter on their chin give them away.
Reading aside, I am certain it leads to attacks of canine bad breath that can produce as many tears in a human as a freshly cut onion.
As far as why from the litter box: I think the litter crumbles add a satisfyingly crunchy coating to it, with a piquant citrus accent. I think they also like to pretend that the litter bin is a take-out box.
As for the question about the box itself, the one we have is a LitterMaid multicat, self-cleaning litter box. It's about seven years old. I understand that for some years in between then and now, they cheapened it a bit, with unsuccessful results, but that they have brought the base quality back to the acceptable range recently. As I said, ours is seven years old, so I don't know the real status of the brand's current quality.
Best wishes,