Just to expand a bit on Jeff's answer for (2), M-theory contains a three-form with a four form curvature. Horava and Witten shower that one could associate the $E_8 \times E_8$ heterotic string with M-theory on $S^1/\mathbb{Z}_2$. The boundaries each have an $E_8$ gauge theory on them. Soon after, Witten used $E_8$ bundles to determine the quantization of the four form in M-theory in hep-th/9609122. This quantization is, interestingly, shifted from being integral. $E_8$ index theory was used spectacularly to compare the partition functions of IIA and M-theory in the ginormous paper of Diaconsecu, Moore and Witten, hep-th/0005090. As Jeff says, the paper of Diaconescu, Moore and Freed is the most modern way of looking at the subject using (shifted) differential cohomology. One of the conclusions of that paper is we just don't know whether the use of $E_8$ bundles for quantizing the M-theory four-form is real or just a convenient trick. But given the other ways $E_8$ seems to be hanging around M-theory (for example, the split real form of $E_8$ shows up when you compactify M-theory on $T^8$), I'd guess the former.
Just to expand a bit on Jeff's answer for (2), M-theory contains a three-form with a four form curvature. Horava and Witten shower that one could associate the $E_8 \times E_8$ heterotic string with M-theory on $S^1/\mathbb{Z}_2$. The boundaries each have an $E_8$ gauge theory on them. Soon after, Witten used $E_8$ bundles to determine the quantization of the four form in M-theory in hep-th/9609122. This quantization is, interestingly, shifted from being integral. $E_8$ index theory was used spectacularly to compare the partition functions of IIA and M-theory in the ginormous paper of Diaconsecu, Moore and Witten, hep-th/0005090. As Jeff says, the paper of Diaconescu, Moore and Freed is the most modern way of looking at the subject using (shifted) differential cohomology. One of the conclusions of that paper is we just don't know whether the use of $E_8$ bundles for quantizing the M-theory four-form is real or just a convenient trick. But given the other ways $E_8$ seems to be hanging around M-theory (for example, the split real form of $E_8$ shows up when you compactify M-theory on $T^8$), I'd guess the former.