You're right, the substitution step is false. Apologies for posting nonsense. Abobe is a new try to deal with the convergence issue, hoppefully correct to the end this time.

Yes, of course. I've added more details. Sorry for being too succinct. It's not easy to give the exactly right amount of hint.

@sajjad veeri: for $n=3$ see my answer to the follow-up post by Moritz Firsching.

@Moritz Firsching: I have now found the exact value of $\operatorname{exp\_abs}(3)$ via an amazing result of Borwein and co-workers, please see above.

The best is a closed form expression. It looks like $I_3=\frac{\pi^2}{6}\, W_3(1,1)$, where $W_3(1,1)=\frac{476}{525}A+\frac{52}{7\,\pi^2}\frac{1}{A}$ with $A=\frac{3}{16}\frac{2^{1/3}}{\pi^4}\Gamma(\frac{1}{3})^6$ is the expected radial distance $\mathbb{E}|X+Y+Z|$ for $X,Y,Z$ uniform on $S^3$, as given in scholarship.claremont.edu/jhm/vol6/iss1/7 (on page 100). (Timothy Budd pointed to this paper in the related MO post.)

(1) Quick question: yes, thanks, corrected (2) I will write you an email (tomorrow, I hope)

Please let me know if anything is unclear or too succinct. I tried to be verbose, but of course it always depends.