This answer is only a bit of a sketch for (2), because it's been a while since I thought/heard about this. And I interpret 'physical considerations' loosely, since this all comes from string theory, whose status as a physical theory is open to debate (at the very least, nothing below comes from experiments!).

There is, in M-theory, a locally-defined 3-form 'higher connection', whose 'curvature'  is a 4-form. This is related to the existence of a bundle 2-gerbe/circle 3-bundle whose characteristic class is this 4-form. Really all that people see of this is the low-energy limit which is 11-dimensional supergravity. Since spacetime $X^{11}$ as conceived in M-theory is 11-dimensional, any classifying map $X^{11} \to K(\mathbb{Z},4)$ for one of these higher structures wouldn't be able to distinguish $K(\mathbb{Z},4)$ from $BE_8$. Thus the bundle 2-gerbe might secretly be an $E_8$-bundle.

Alternatively, in regular string theory one has the $H$-flux, which is 3-form, and is commonly understood to be the curvature 3-form associated to a bundle gerbe/circle 2-bundle. However exactly the same argument as before means that perhaps what is going on is that there is an $\Omega E_8$-bundle instead of a bundle gerbe (this is the point of view espoused to me by Jarah Evslin a few years back). Due to the heterotic string theory $E_8\times E_8$, and various models with compactifications, it is not wholly unreasonable to expect $\Omega E_8$ to turn up at some point.

As far as (1) goes, which version of $E_8$ are you after? Assuming it is a compact real Lie group, it is 2-connected and we have $\pi_3(E_8) = \mathbb{Z}$. After that I'm not sure how the homotopy groups are calculated. Morse theory?