So octonion set provides the largest normed division algebra, and starting with sedenions, Cayley-Dickson construction provides algebras with zero divisors.
From what I understand, it means there are pairs of non-null sedenion $(s_a,s_b)$ for which $ s_a·s_b = s_0$ is correct – where $s_0$ stands for the null sedenion $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$. But maybe I already misunderstood something here?
But given that higher order algebra gradually lose algebraic properties as the scale of dimension is climbed, are they any pair of non-null sedenion $(s_c, s_d)$ for which $s_c/s_0=s_d$ holds? That is, is it valid, at least in some cases, to divide by the null sedenion?
No it is not. In the Sedenion algebra you have not only one couple but many such that their product is 0, and 0 is still absorbant for the original product. So you cannot "divide" by 0 because there is no exclusivity.