Some special properties of dimension 8, in addition to the ones you identify:

 - [Bernstein's problem][1] 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. 


[![Coxeter diagrams of Euclidean reflection groups][2]][2]

 

 - There are 4 infinite families of [holonomy groups of Riemannian][3] manifolds, with two exceptional cases of $G_2$ and $Spin(7)$, the latter being in dimension 8. 

 - As pointed out by @YCor, [triality][4] 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][5] 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). 


  [1]: https://en.wikipedia.org/wiki/Bernstein%27s_problem
  [2]: https://i.sstatic.net/fIywC.png
  [3]: https://en.wikipedia.org/wiki/Holonomy#The_Berger_classification
  [4]: https://en.wikipedia.org/wiki/Triality
  [5]: https://projecteuclid.org/euclid.gt/1510858992