So in order to actually assemble these cookies, you need to select the right recipe. You need cookies that will maintain their shape when baking and that won't rise too much. I used this one and it worked out pretty well.
When making the cookies roll the dough out to a little under .25 inches thick. I measured mine with my calipers to about .22 inches for most of them. When you assemble the cookies, you might want to cut the slots with a serrated knife like a steak knife or a paring knife to open then up a little bit. Just shave down the edges so that the cookies fit together nicely. Also, most of the shapes can just slide together, but for the reindeer I needed to use icing to "glue" it all together.