The monochromatic slices of a finite $X$ are non-connective spectra, so $X$ surely is not their wedge sum. You might find Mark Hovey's paper on Hopkins' chromatic splitting conjecture interesting: that conjecture would imply that for finite $X$ the product of the $L_{K(n)}X$ contains $X_p$ as a summand. [I think the original formulation of that conjecture is now known to be false, though.]

No, that must be false. I don't even see how you'd get a map between $X$ and the (wedge of the) $M_nX$. Here $M_nX$ = fiber $L_nX \rightarrow L_{n-1}X$ is what I'd call the monochromatic slice.

Isn't this pretty immediate? by assumption the map $X\rightarrow\ast$ is a $K(n)$-equivalence for all $n$. Then it is also an $E$-equivalence for the infinite wedge $E=\bigvee K(n)$. And since $X$ is $E$-local, the map is an honest equivalence.

Another good way to "cheat" is to compare the ASS and the ANSS in that range. This is done in Ravenel's green book, chapter 4, pg. 143-144 (2nd edition). The book is available online: math.rochester.edu/u/faculty/doug/mu.html

Your claim "$G$ has the same homotopy type as a product of odd-dimensional spheres" is only true rationally, isn't it?

See Theorem (2.2) in arxiv.org/abs/dg-ga/9506004

Supporting Stopple's comment, I'd recommend Hirzebruch's 2007 Euler lecture, which deals with Euler's divergent series summations at the beginning: mathnet.ru/php/…

I haven't looked at this before, but at least the page hamilton.nuigalway.ie/Hap/www refers explicitly to infinite groups: "HAP is a homological algebra library for use with the GAP computer algebra system, and is still under development. Its initial focus is on computations related to the cohomology of groups. Both finite and infinite groups are handled, with emphasis on integral coefficients." So this seems very relevant to the original question.

A related question is this: mathoverflow.net/questions/16474/matrix-representation-for-f‌​-4/… (I would link to Marty's comment on his own answer if I knew how to do this.)