Finding the asymptotic growth of the expected number of extreme points of the quadrant hull in $d$ dimensions seems to be a well studied problem. Interestingly, the solution appears to vary widely by the distribution. As previous posts point out (Average size of extreme points of convex hull of $N$ points), the solution for drawing uniformly from convex bodies is $O(log^{d-1}(n))$, whereas from a ball it is $O(n^{\frac{d-1}{d+1}})$.
The unbounded case has received attention as well, particularly for spherically symmetric distributions (For instance: https://www.jstor.org/stable/3214867?seq=1#page_scan_tab_contents).
I am wondering about the case morally in the middle--drawing random points from a halfplane. I have not been able to find any literature on this: what is the expected number of extremal points in the quadrant hull of $n$ points drawn uniformly from a halfplane $H_d$?