Why is E6 favored over E8 for GUT building? What is the simplest explanation on why is E6 gauge group more favored as a group for Grand unified theory builders, while E8 is not? What about other exceptional groups? Which of them originate from the string theory in the most natural way, and why?
 A: Saying that $E_6$ is "favored" over $E_8$ in GUT model building is a big understatement.
There can't be any grand unified theory with an $E_8$ gauge group because $E_8$ has no complex representations, i.e. representations that are inequivalent from their complex conjugates. The existence of complex representations is a necessary condition for the theory to contain chiral fermions, i.e. Dirac fermions whose left-handed components carry different quantum numbers (and interactions) than the right-handed components. One may also say that complex representations are needed for the violation of C, P, and CP.
$E_6$ is the only one among five exceptional groups that has any complex representations. It's related to the fundamental ${\bf 27}$ or antifundamental $\overline{{\bf 27}}$ representation of the group which are interchanged by an outer automorphism of $E_6$, a symmetry that boils down to the ${\mathbb Z}_2$ symmetry of its Dynkin diagram. $E_6$ is the only exceptional Lie group with a nontrivial symmetry of the Dynkin diagram.
All other exceptional Lie groups, namely $G_2, F_4, E_7, E_8$, only have real representations, a fact that can be seen by looking at their real fundamental representations, too. The spectrum of gauge theories using these groups would be inevitably left-right symmetric, and therefore experimentally excluded. Patterns about particles such as "neutrinos have to be left-handed" would be impossible.
Despite the comments above, $E_6$ is a subgroup of $E_8$. So in string theory, it is possible to break $E_8$ (a key group e.g. in heterotic string theory) to $E_6$ by stringy effects, e.g. by nontrivial configurations of the $E_8$ gauge field as a function of the extra (compact) dimensions in $E_8\times E_8$ heterotic string theory. Spontaneous breaking of $E_8$ by field-theoretical methods (Higgs mechanism) is no good because it would only produce real representations of $E_6$ again. In string theory, $E_6$, a viable GUT group, may emerge as a subgroup of $E_8$ (or $E_7$). $G_2$ and $F_4$ are too small to be relevant for GUT model building.
All papers that claim to build viable models of particle physics from an $E_8$ field theory are pseudoscientific gibberish, denying elementary features of the groups in which the known quantum fields transform. In the case of Garrett Lisi's paper, the absence of complex representations is the main point of the paper by Garibaldi and Distler.
