We currently have three cats in the house, using two Outdoor Litterboxes. I usually clean twice a week, although one of the boxes is less used and probably doesn't need cleaning that often. The boxes do open from the outside and cleaning, for me, consists of a couple of minutes of scooping with a larger "hand shovel" with slots and holes in it. Since the boxes are larger and deeper than a typical litter box, there's more litter for the cats to use. That does mean there's more to scoop, of course... but it can be done less frequently. The final version will be easily detachable from the tunnel by the owner so you could unhook it, dump the contents, and refill it with fresh litter rather than shoveling/scooping.
Addressing your question about your dog... it's difficult to find the corrugated accordion plastic in larger sizes. It's nearly impossible to find it in a rectangular shape which would adapt for your use. It would be tough for me without seeing it to come up with a good alternative since that might be affected by the size of the door and dog, as well as the height of the door above grade. The solution probably depends on how much of your problem is due to wind or air flow vs. the simple fact that it's cold outside.
I would probably build some sort of tunnel or wind break around the exterior of the pet door using stacked bricks and/or wood (or Trex) and see if that reduces the problem. Obviously, you have to be careful that it's sturdy enough that the dog can't just bump it and cause it to collapse. If it's secure enough, though, that'll give you some protection from the air flow outside. Before that, I'd probably try to find a pretty heavy-duty box and cut a hole in it to use as a trial run just to see if it helps. You might need to "weatherize" it somehow... or just use a plastic box or plastic panels that are adapted. I'm seeing more double-paned, insulated pet doors these days, too. Some of them look like they'd be much better than the typical pet door in terms of insulation.
I hope that helps. Thanks for the encouragement regarding The Outdoor Litterbox.
David
http://www.outdoorlitterbox.com