The number 1558 appears in the URL of the question. But note that questions aren't necessarily numbered consecutively (but increasingly). These numbers probably are some internal id.

When studying the algebraic properties of MZV (eg. in "Algebraic Aspects of Multiple Zeta Values", math/0309425) Michael Hoffman indeed calls treating \zeta(1) as Euler's Gamma a "happy choice" (after Theorem 3.5).

You don't think that 1+1+1+... = -1/2 has some decency? (Of course, that's \zeta(0).)

Looking at the paper I found the typo: c_n = n! 4^{-n} ... Also, your quite right; the integral does have a nice closed form coming from writing it as a Selberg integral. I put details into a new answer.

Thanks a lot for working on my intuition! Maybe I'm misreading your final statement but I think you need to assume the map between G and G' to exist from the beginning. Take eg. G=Z/4, G'=Z/2xZ/2 and factor out by Z/2 subgroups.

No, I'm not saying that. (9,12) = 3 (3,4) and, as you and javier observed, {(3,4),(1,1)} is a basis of Z^2. Therefore, your quotient is isomorphic to Z^2 / (3,0) which, of course, is Z_3 x Z.