Let me first answer the relation between string theory and $E(8)$ (I don't think I can answer the rest.). A common appearance of $E(8)$ in strings theory, is in the gauge group of Type HE string theory, i.e., in $E(8)\otimes E(8)$. Now, this appears in Type HE string theory because of the fact that it is an even, unimodular lattice. But, it is interesting, for another reason; due to the embedding of the Standard Model Subgroup:
$$SU(3)\otimes SU(2)\otimes U(1)\subset SU(5)\subset SO(10)\subset E(6)\subset E(7)\subset E(8)$$
(Admittedly, this is a very common sequence to find in many string theory books (and old papers,.). )
That's a lot of embeddings, but notice! The first group here, in the Standard Model subgroup, the second, third, fourth, fifth, are GUT subgroups. And $E(8)$ happens to be the "largest" and "most complicated" of the exceptional lie groups. So a TOE better deal with $E(8)$, somewhere!
I don't know about the relation between monstrous moonshine and string theory, but Wikipedia knows a bit (or in this case, perhaps a lot). .
There is definitely a connection with number theory, and more than just that. . And even more: .
Not joking! EM is the curvature of the $U(1)$ bundle . Weak is the curvature of the $SU(2)$ bundles. Strong is the curvature of the $SU(3)$ bundled ; . Gravity is the curvature of spacetime . I.e. 1D manifold, 2D, 3D, 4D $\implies$ 10 D .
I will also put Ron Maimon's commennt here, as comments are on the kill-list of SE:
There is another point, that E(8)
is has embedded E6xSU(3), and on a Calabi Yau, the SU(3) is the holonomy, so you can easily and naturally break the E8 to E6. This idea appears in Candelas Horowitz Strominger Witten in 1985, right after Heterotic strings and it is still the easiest way to get the MSSM. The biggest obstacle is to get rid of the MS part--- you need a SUSY breaking at high energy that won't wreck the CC or produce a runaway Higgs mass, since it seems right now there is no low-energy SUSY.