What you call the derived polygon is a construction that has appeared many times in literature. I believe the first occurrence was in a paper by Darboux, "Sur un problème de géométrie élémentaire", but it has also been a topic of many problems and articles in the American Mathematical Monthly, which is where I've learned about them. :-)

Not only is the limiting shape a polygon with opposite sides parallel, but it is in fact the affine image of a regular $n$-gon (phrased this way you don't have to restrict to even number of sides). The proof boils down to the fact that every polygon is a linear combination of regular polygons, and these are eigenvectors of the associated circulant matrix. This is nicely explained in a monthly article, "A Polygon Problem", by E. R. Berlekamp, E. N. Gilbert and F. W. Sinden.

Another interesting perspective is to think of the limiting shape as equally distributed points on an ellipse. This, in fact, makes an analogy with the continuous version of the problem. If you have a curve in $\mathbb R^2$ and want to consider a second order parabolic evolution equation which is invariant under affine transformations, you will be studying a flow called the affine normal flow. In "Contraction of convex hypersurfaces by their affine normal", Ben Andrews proves that the affine normal flow contracts every convex hypersurface to a point, and if you scale the limiting shape accordingly, the shape converges to an ellipsoid. In fact the midpoint iteration from before is just the discrete analog of this evolution for 2D curves!

Now, coming to your question about the 3 dimensional polytopes. Your discrete evolution is clearly invariant under affine transformations and should approximate a second order evolution equation, which would then coincide with the affine normal flow. Since in dimensions higher than two, the mesh becomes more refined in every iteration, the limiting surface should in fact converge to the ellipsoid, instead of just a discrete approximation of it as in the 2D case. Unfortunately I do not know enough about numerical analysis or differential geometry to tackle this problem, and just gave a heuristic answer. If you add those tags perhaps an expert will explain how to prove a result like this.

What you call the derived polygon is a construction that has appeared many times in literature. I believe the first occurrence was in a 1878 paper by Darboux, "Sur un problème de géométrie élémentaire", but it has also been a topic of many problems and articles in the American Mathematical Monthly, which is where I've learned about it. :-)

Not only is the limiting shape a polygon with opposite sides parallel, but it is in fact the affine image of a regular $n$-gon (phrased this way you don't have to restrict to even number of sides). The proof boils down to the fact that every polygon is a linear combination of regular polygons, and these are eigenvectors of the associated circulant matrix. One, then observes that the limiting shape is determined by looking at the two largest eigenvalues. This is nicely explained in a monthly article, "A Polygon Problem", by E. R. Berlekamp, E. N. Gilbert and F. W. Sinden.

Another interesting perspective is to think of the limiting shape as equally distributed points on an ellipse. This, in fact, makes an analogy with the continuous version of the problem. If you have a curve in $\mathbb R^2$ and want to consider a second order parabolic evolution equation which is invariant under affine transformations, you will be studying a flow called the affine normal flow. In "Contraction of convex hypersurfaces by their affine normal", Ben Andrews proves that the affine normal flow contracts every convex hypersurface to a point, and if you scale the limiting shape accordingly, the shape converges to an ellipsoid. In fact the midpoint iteration from before is just the discrete analog of this evolution for 2D curves!

Now, coming to your question about the 3 dimensional polytopes. Your discrete evolution is clearly invariant under affine transformations and should approximate a second order evolution equation, which would then coincide with the affine normal flow. Since in dimensions higher than two, the mesh becomes more refined in every iteration, the limiting surface should in fact converge to the ellipsoid, instead of just a discrete approximation of it as in the 2D case. Unfortunately I do not know enough about numerical analysis or differential geometry to tackle this problem, and just gave a heuristic answer. If you add those tags perhaps an expert will explain how to prove a result like this.