I think I'm logged in, but if not, this is GD again.
As to spheres vs algebra, years ago I had a confab with Baez in which I posited the idea that octonion maths is in a sense holographic, as most of the bits and pieces we like can be used to derive the remaining bits and pieces. That is, no given bit is any more fundamental than any other. Parallelizable spheres yield division algebras; and visa versa. As you do, I start with algebra, but in my heart I view algebra as a kind of intellectual microscope, devised by humans to better see structures that exist without us. I mean, in the total absence of any intellect in the universe, planets and stars still take roughly spherical form. However, without intellect algebra does not exist. But again, I start with algebra, because it allows me to play with things more fundamental.
You say: "Then there is that pesky issue of whether or not two algebras with demonstrably different structure can be truly considered isomorphic." I assume you are referring to the left/right octonion thing. If so, isomorphism is demonstrable. I think I even did it in my windmill book. There is only one octonion algebra, but many ways to organize it. Really, one needn't use integers to to label the units. One could use fruit, or puppies, or anime characters. Integers just allows us to see some structures that are otherwise hidden. Like my fav multiplication table with e1 e2 = e4, imposing invariance with respect to cycling and doubling of integer subscripts (yielding a finite invariance group of order 21).
Anyhum, give me any two multiplication tables for O, and I'll build an isomorphism. None of the best minds in the fields (like Conway and Sloane and others) ever discuss two versions of O that are not isomorphic. There is one O. Until proven otherwise, of course.
