I don't have a good intuition for the following problem:
Given three random (unit) vectors $a, b, n \in \mathbb{R}^d$, chosen u.a.r. as points on a unit sphere, with $n$ being the normal vector of a hyperplane $h_n$.
As the dimension $d$ increased, does the probability of $h_n$ separating $a$ from $b$ increase, decrease or stays the same?
It stays the same. Choose first the hyperplane $H$ randomly. The probability of $a\in H$ or $b\in H$ is $0$. The hyperplane separates the sphere in two hemisphers so in particular the probability of $a$ being in one of them is $1/2$ and the same is true for $b$. So essentially we have two independent bernoulli random variables.