Well I just cut out that image I linked to and it doesn’t remotely approximate a globe when cut out and curved. Uncle.
I can’t quite figure out what you’re trying to do. When I think of clay, I imagine nice plastic modeling clay, which is pretty easy to mold into whatever shape you need. I would think that just about any reasonable pattern would let you cover the sphere mostly-uniformly, after which you just need to roll it around a bit to smooth the edges and overlaps. You need that last step in any case, as long as you stick to cutting things out of flat pieces of clay. Why does that not work? How about making a hollow mold of the desired size, and just squeezing it around the inner sphere?
Covering a sphere with paper is a harder proposition. Paper is not very elastic, so it’s hard to cover a large section of the sphere without the edges crinkling. This problem gets worse approximately as the square of the radius of a circle inscribed within the sheet of paper, so narrow pieces work better than wide ones. This is the point of the orange-peel unwrappings like the 12-gore model you linked earlier. The other projections you linked (like the Mollweide) are not designed for use in covering a physical globe; they are just flat maps that preserve some desired properties of the sphere (such as preserving lines of constant bearing, minimizing distortions, or keeping landmasses contiguous).
Ok, I think we’re on the wrong track, here. To wrap a 1-inch or smaller sphere with mooshable clay, you just need a ribbon of clay with the edges cut in a zigzag. Wrap around the bead once, and squeeze the points together with your fingers. At that point, you can roll it around between wetted palms, or smooth it out by applying a thin mud with a brush. How neat do they have to be? As the lady selling Waterford crystal told me, “These are handmade, which means they won’t be perfect.”
So, how to get the zigzags? Pinking shears sprang to mind first. Then I thought about kitchen devices. There’s a ziggy cutter wheel for pasta, and a similar thing for the edges of pies.