Well -- everyone's got their own taste. But it's not really "ad hoc". Constructions via Jordan algebras lead to a deep understanding of Lie algebras with minuscule parabolics. Constructions (like above) via structurable algebras lead to a deep understanding of Lie algebras with 2-step parabolics. Since all simple Lie algebras have a natural 2-step parabolic (Heisenberg parabolic), this is important in practice!

I think so, since the $\alpha^\vee$ are linearly independent (over $Q$)... I'm taking simple roots. Or am I misunderstanding or making a dumb mistake?

I've updated my answer, which might explain the connection to $SO_{2n+1}$-factors.

Oh -- you beat me to it! Well, I'm glad that our answers are equivalent.

Well, $\nu_2(n+13) = \nu_2((n+1) + 12)$. And 12 is special :)

I'm not sure -- I'd guess that people like Opdam and Dyson are aware of the fact that the modern MacDonald-conjecture style formulas specialize to Selberg's integrals, which are equivalent in a special case to Dixon's sum. Maybe ask Opdam directly?

The evaluation of this sum was treated by Dixon, "On the sum of the cubes of the coefficients in a certain expansion by the binomial theorem" in Messenger of Math. 20, 79 (1891)! See also en.wikipedia.org/wiki/Dixon%27s_identity.