I remember something about "more later" in my last post (but did not reread it prior to typing this new post). So I will share the second and third (pretty much "only other") details of my discworld:
1. Time in discworld is divided into 8 hours, each equal to 3 of our hours (I knew I could type our hours in a sentence one day). Why 8 and not 24 or 12 or some other number that makes sense in sphere land? Because I figure if I were dividing a disc into pieces, I would cut it in halves. A pie chart divided into 12 or 24 pieces would require cuts along radii rather than along diameters. Since a few cuts all the way through are easier than spacing out cuts along the radii, the choice was 2, 4, 8, 16, or 32 hours and 8 seemed like enough without being too much (it also conveniently factors into 24 evenly). Fitting the easy to cut theme, each of the 8 hours are divided into quarters (and thus are 45 minutes) and each of the quarters are themselves divided into squares (which are 11.25 minutes long). Then you have seconds, which are a quarter of a square (the name comes from 1/32's, which is what this would be; they are about 3 minutes long). A second is finally divided into 100 counts (each about 1.7 seconds long). There is rarely any need to tell time accurately beyond hour and quarter. This is a medieval-esque society, not the modern world.
2. There are three views concerning the underside of the disc, keeping in mind that the inhabitants of the disc know their world is flat and that they inhabit the top side of the disc (which is not to say all inhabitants know this; just that it is known to be fact amongst the learned and the few heroic souls who have ventured far enough). The first is that the bottom of the disc is of no concern. As a matter of simplicity, adherents to this view believe the disc is flat earth on the underside and only the top side exhibits the mountains, valleys, forests, and oceans with which they are familiar. In essence, the disc is a one side affair. A small subsection of this school of thought holds that the underside is not exactly barren but entertains such an alien geography that exploration would be useless (or, at the very least, would not be worth overcoming the lack of gravity holding you to the ground; I mean, it is the underside of a disc floating through space, or so they imagine).
The second school of thought holds that the disc is "featured" on both sides. This mountain has brothers on the bottom, this ocean its counterparts. Within this school are those who believe that a mirror image matches up directly with the perceived world (such that a mountain's twin extends downward directly beneath the peak which extends upward) and those who believe that while both sides are filled with mountains and oceans and forests, they are arranged differently (making the bottom of the disc a new world to explore). Adherents of the former view believe excavation at the deepest parts of the earth (ocean bottom, canyon floor, deep valleys, etc) represents the best chance of breaking through to the other side.
The third view holds that the mountains and valleys of the disc are formed from deformations through the disc. This resembles the second school in that there is terrain on the underside but differs in that the mountain on this side is created from a mountain-shaped valley on the underside. Likewise a wide valley on this side is matched by a wide hill on the underside. Under this view, a deep enough cave further extended with mine shafts might break through the layer of the disc into the valley of the underside. Considering the link between geographies, what must this school imagine about plant and animal life below?