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:

$$SU(3)\otimes SU(2)\otimes U(1)\subset SU(5)\subset SO(10)\subset E(6)\subset E(7)\subset E(8)$$

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 you can refer to Wikipedia.

There is definitely a connection with number theory. And even more: .

$$1+2+3+4=10$$

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 $SU(3)$ bundle. Gravity is the curvature of spacetime . I.e. 1D manifold, 2D, 3D, 4D $\implies$ 10 D .