The general situation, where CH fails, may be informed by
the Keisler-Shelah isomorphism
theorem, which
asserts that two first-order structures have isomorphic
ultrapowers if and only if they have the same first-order
theory.
In particular, for any infinite group $G$ at all, of any size, we
may take a countable elementary subgroup $H$, meaning in
particular that they have the same first-order theory, and
so there is a nonprincipal ultrafilter $U$ on an index set $I$
such that the ultrapowers $G^I/U\cong H^I/U$ are isomorphic. Since every first-order structure maps
elementarily into its ultrapowers, this means in particular
that $G$ maps elementarily (and hence monomorphically) into
an ultrapower of $H$, a countable group.
Thus, this fully answers the version of question 2 in which
we allow the ultrafilter to live on a bigger index set:
Theorem. For every group $G$ there is a countable group $H$ and a free ultrafilter $U$ on a set, such that $G$ embeds into the ultrapower $H^I/U$.
If you want to insist that the ultrafilter
concentrate on index set $\mathbb{N}$, however, then things become more complicated. If the CH holds, then the
Keisler-Shelah theorem shows that any two groups of size at
most $2^{\aleph_0}$ and with the same theory have
isomorphic ultrapowers by an ultrafilter on $\aleph_0$, and
so the desired result is attained. In the non-CH case,
however, what we seem to get is that for any cardinal
$\lambda$, if $\beta$ is smallest such that
$\lambda^\beta\gt\lambda$, then any two groups of size
$\beta$ with the same theory have isomorphic utrapowers
using an ultrafilter on $\lambda$. Thus, they each map into
an ultrapower of the other.
The Keisler-Shelah theorem was proved first by Keisler in
the case that GCH holds, using saturation ideas as in
Simon's answer. The need for the GCH was later removed by
Shelah.
I can't give a comprehensive history (if you don't get that here, you might try [hsm.se]---a lot of mathematicians are active on that site), nor can I explain how or why the theory of spin manifolds first emerged. But I think I can say something about how and why spin manifolds became important.
The pre-history is an observation due to some combination of Atiyah and Hirzebruch that a certain characteristic number, the $\hat{A}$-genus, happens to take integer values on spin manifolds (a priori it is a rational number). I believe they were able to prove it using spin cobordism theory (though I'm not sure), but it still called for a convincing conceptual explanation.
This was certainly on Atiyah and Singer's minds when they were working on the index theorem, which computes the Fredholm index of an elliptic (pseudo)differential operator in terms of topological data. This is undoubtedly why in their Index of Elliptic Operators III they introduced the spinor Dirac operator associated to a spin manifold. The index of this operator is obviously an integer on one hand, and on the other hand they calculated that the index is precisely the $\hat{A}$-genus, beautifully explaining Atiyah and Hirzebruch's observation.
Moreover, Atiyah and Singer realized that many of the other operators used to give applications of their index theorem (including the de Rham operator, the signature operator, and the Dolbeault operator) can be constructed in a uniform way using the representation theory of Clifford algebras, so in a certain sense the spinor Dirac operator is the fundamental example in index theory and therefore has its tentacles in many different parts of geometry and topology. This observation explains why spin geometry is so ubiquitous in the theory of positive scalar curvature obstructions, for instance.
The significance of this operator, and therefore spin geometry, was elevated and clarified by the development of K-homology (the homology theory corresponding to the K-theory spectrum). Perhaps the most fundamental mathematical explanation of the importance of spin manifolds is that the spin condition corresponds to orientability for the KO-theory spectrum, and spin manifolds equipped with spinor Dirac operators correspond to the fundamental classes. (The counterpart for traditional K-theory is the spin$^c$ condition.) This was inspired by Atiyah, who argued that there should be a model of K-homology in which the generators are elliptic pseudo-differential operators, and sorted out by Baum and Douglas. The result is that spin geometry infiltrates many problems that involve topological K-theory.
The next (and current) chapter in the story involves the recent interest in loop spaces of manifolds, inspired by physics. It turns out that a spin structure on a manifold is in some sense the same thing as an orientation on its loop space, an observation which Witten used to sketch a proof of the Atiyah-Singer index theorem. There is another kind of structure on a manifold - a string structure - which corresponds to a spin structure on the loop space, and Witten constructed an invariant (the Witten genus) which he argued ought to be the index of a loop space Dirac operator. So far as I know nobody knows how to construct such an operator, but the Witten genus provided motivation for a lot of exciting modern geometry and topology, including topological modular forms and Stolz-Teichner functorial field theories.
Best Answer
Here's what Dan Voiculescu himself gave as motivation: