Addressing the moments in dimensions greater than 2, if $A_d$ is the area of a triangle whose vertices are chosen uniformly at random from a unit sphere in $\mathbb{R}^d$, the even moments I’ve computed with symbolic integration for $d=3$ all satisfy:
$$E(A_3^{2m})=\frac{3 (4 m+2) (m!)^3 (6 m+1)!}{(3 m)! (2 m+1)! (4 m+3)!}$$
I’ve checked this up to $m=15$. The first 10 values are:
$$\frac{1}{2},\frac{13}{30},\frac{323}{700},\frac{23}{42},\frac{4495}{6468},\frac{33263}{36036},\frac{65231}{51480},\frac{21460999}{12033450},\frac{30741431}{12009140},\frac{8965109}{2401828}$$