What is special to dimension 8? Dimension $8$ seems special, as the partial list below might indicate.
Is there any overarching reason that dim-$8$ is "more special" than, say, dim-$9$?
Surely it isn't it, in the end, simply because $8=2^3$, but $9=3^2$? Or that $\phi(8)=4$ but $\phi(9)=6$?


*

*The sphere packing problem in dimension
$8$: arXiv.
Annals Journal.
Quanta article.

*Octonians (Wikipedia).
John Baez: The Octonions.

*E8. See also Garrett Lisi's $E_8$ Theory (Wikipedia). 

*Wolchover, N. "The Peculiar Math That Could Underlie the Laws of Nature." Quanta magazine (2018).

*De Giorgi's conjecture: 
Abstract: "A counterexample for $N\ge9$ has long been believed to exist. ...we prove a counterexample [...] for $N\ge9$."
Del Pino, Manuel, Michal Kowalczyk, and Juncheng Wei. "Annals of Mathematics (2011): 1485-1569.
DP,MKM,JW. "On De Giorgi’s conjecture and beyond." PNAS 109, no. 18 (2012): 6845-6850.

*Both the Snake-in-a-Box and the Coil-in-a-box problems have been solved
for $d \le 8$: arXiv abs.
For $d>8$, only lower bounds are known.

*Bott Periodicity:
"period-$8$ phenomena"
(as per @Meow's comment).

*The Simons minimal cone,
a $7$-dimensional cone in $\mathbb{R}^8$
(as per @DeanYang's comment).
 A: Some special properties of dimension 8, in addition to the ones you identify:


*

*Bernstein's problem holds up to dimension $n=8$. The only function of   $\mathbb{R}^{n-1}$  whose graph in $\mathbb{R}^n$ is minimal is a linear function. This fails in dimension $n=9$, with failure due to the existence of the Simons cone in dimension 8, so it's related to your last bullet point. 

*There are 4 infinite families of Euclidean reflection groups, with exceptional ones only up to dimension 8. This is related to the existence of the exceptional simplex reflection groups and exceptional Lie algebras. 



*

*There are 4 infinite families of holonomy groups of Riemannian manifolds, with two exceptional cases of $G_2$ and $Spin(7)$, the latter being in dimension 8. 

*As pointed out by @YCor, triality holds for $Spin(8)$. $Spin(8)$ has three 8-dimensional irreducible representations which are permuted by the $S_3$ action associated with the symmetries of the $D_4$ Dynkin diagram. 

*Cohn and Kumar found various tight simplices including a maximal 15 point tight simplex in $\mathbb{HP}^2$ which is 8 dimensional. A simplex in this case refers to a collection of equidistant points. 
There are several other examples in the comments of phenomena where 8 dimensions is the first dimension in which the phenomenon appears (or is known to appear), but I've listed examples that seem to be special to dimension 8 (and most seem to be connected to the phenomena that you've already identified). 
