As I understand, in Geometric Algebra (GA) the dual of a blade is the rest of the space, ie: what you have to multiply a blade by to get to the top blade (the pseudoscalar). Thus in 3D, meaning G(3,0,0), the dual of e12 is e3. The dual of a blade is usually obtained by post-multiplying by the pseudoscalar, but in PGA, meaning G(n,0,1), this doesn’t work because the pseudoscalar is (for n=3) e0123 and the e0 in this (the null direction) will multiply with any blade with another 0-index in it to produce 0. Although this seems problematic, it doesn’t really matter. You don’t need the pseudoscalar multiplication to define the dual, which is perfectly well defined as being the rest of the space. So the reason the coefficients reverse order (in the code by enki that you quote) is just a consequence of listing the terms in this order {1, e0, e1, e2, e01, e02, e12, e012} : you see that the dual of blade i here is blade 7-i, for i=0 to 7.
Your second question goes to the heart of the PGA innovation (and controversy). Hopefully complementary to what Leo has said above, I would add that yes, at first it is unpleasant to find the wedge product reducing the dimension of geometrical objects, as you say. But nothing actually changed from regular GA’s. The wedge still increases the blade dimension, but because, in PGA, Euclidean objects reduce in dimension as the grade of the blades representing them increases, the strangeness you point out results.
Personally, I found it quite enlightening to slog through the long debate here (so long I could not get Firefox to print it):
Much of it is concerned with Eric Lengyal’s claims here:
http://terathon.com/blog/projective-geometric-algebra-done-right/
Like you, Lengyal was perturbed by the use of the “dual algebra” for Euclidean objects in PGA, and he goes on to dualise the operators instead of the objects. This produces all the same computations. I was satisfied by their debate and, concluded that there are two equivalent ways to talk about it. Whichever you prefer (and I do prefer Gunn’s, although it is not yet available as a wall poster on amazon like Lengyal’s: Amazon.com), if you understand what they debate about above, you will understand much more about your question. It is already weird enough to be in projective spaces, so hey, why not reverse the dimensional significance of grade? It recasts all previous projective geometry in an interesting new light. My view is that this is deep magic. In practical terms, it works, and it is, remarkably, a relatively new and powerful advance in Euclidean Geometry. Such things appear, in my calculation, about once every 1000 years.
It would be nice to see this all fleshed out properly in a tutorial paper, as I agree with you that the “Course Notes” can be a bit technical in places for the novice. I found Chris Doran’s remarks on the issues you touch on to be also instructive:
http://geometry.mrao.cam.ac.uk/2020/06/euclidean-geometry-and-geometric-algebra/
And I think this forum can help quite a bit in this respect. Certainly it has helped me. - Tony