I once read a great little article in an SGI’s now-defunct “Iris Universe” which basically said, “just try your code and negate as needed”. You can sit and think and scribble for a long time, trying to figure out whether a particular rotation will go clockwise or counter-clockwise, or whether the culling sense is right. This article (author forgotten) pointed out that, once you have the underlying math right, most of these decisions are boolean; it’s either one way or the other, and the difference is just a minus sign. Rather than spending time with paper and pencil and figuring out exactly what the ordering should be, the author said, “just try it”. Having this permission to do so has saved me hours of worrying about sign problems while programming. The ultimate goal, for me, is a picture on the screen; if a minus sign needs to be added here or there along the way, no problem.

You mention the wedge product (aka the outer product), from geometric algebra. I can appreciate the elegance of GA, but can’t say I understand much of it beyond a hand-waving level. It has an clean way of manipulating various objects and computing intersections between them (though given that GA can’t fully represent a triangle, I see its use as somewhat limited for computer graphics). I’m hoping someone someday writes a book on the topic that doesn’t immediately double back on itself with terminology. I was disappointed to find the promising-sounding “Geometric Algebra for Computer Science” book (sample chapter here) assumes the reader already knows about all there is to know about homogeneous coordinate space and projection, quaternions, Pluecker coordinates, etc. I suspect you’d understand it, but this book (and others on GA) have so far left me in the dust. “Geometric Algebra for Dummies” has yet to be published, unfortunately.

Right-handed coordinate systems are preferred because the cross product was chosen to be right-handed at some point in time. This choice was arbitrary — each component of the cross product could be negated, and everything would still be algebraically consistent. The universe has no preference. The reason a choice has to be made lies in the fact that the cross product is really an abuse of a coincidence that occurs in 3D space for a more general product called the wedge product. (And this abuse forced physicists to make an unnecessary distinction between what they call polar vectors and pseudovectors.) The wedge product between two vectors produces something called a bivector, and it so happens that in 3D, a bivector has 3 algebraically independent components whose magnitudes are exactly those of the components of the cross product. (In 4D, a bivector has 6 components, and in 2D, a bivector only has 1 component.) The tensor representation of a bivector is an antisymmetric matrix containing both positive and negative versions of each cross-product component, so no sign choice needs to be made. It’s only when we want to represent a bivector with 3 numbers instead of the 6 nonzero entries of the tensor that we need to decide on RH or LH.