@Joonas You had better specify that $n>1$ then. By the way: there is of course a common name for these things, but it's cumbersome: left cosets of (non-trivial) cyclic subgroups.

Any singleton is a geodesic in your sense (if you don't impose maximality), which makes problem 2. trivial.

Sorry Jan, I was being sloppy. Of course, an arbitrary element of $\mathbb{Q}$ has no way of acting on the class groups. But the maximal order in $\mathbb{Z}[\Gamma]$ does act on the coprime-to-$|\Gamma|$ part of the class group. I have expanded the answer to explain this.

Note: "all irreducible characters real valued" $\neq$ "all irreducible representations defined over $\mathbb{R}$". A symplectic character does not contribute to the involution count.

Here, I was taking $F_i$ to be Galois over $K$. But in fact, you can extract from this general formulation the heuristic for non-Galois sub-extensions by taking suitable idempotents $e$.

@René and Daniel: Actually, already the fact that $H^1(K,E)$ has elements of arbitrarily large order is enough for the OP's purposes. Identifying $E$ with a group of automorphisms of $E$ as an algebraic variety, given by translation by points, identifies $H^1(K,E)$ with a set of forms of $E$, i.e. curves that are isomorphic to $E$ over $\bar{K}$ as algebraic curves. Now, if a class in this cohomology group is trivialised by a finite Galois extension $F/K$ of degree $n$, then it has order dividing $n$. No need to talk about coverings and "period divides index", I think.

@Paul: if you don't need motivation for the global Galois group, then the local ones are easy to motivate. They are very important (conjugacy classes of) subgroups of the global Galois group. The first thing one tries almost always when trying to solve a global question in number theory is to reduce it to its counterpart "locally everywhere". Sometimes, such a reduction works and one is happy, and sometimes it doesn't work, and then understanding the failure of this is also interesting.

The image can be characterised very explicitly: it consists of all those automorphisms that act as an integer power of Frobenius on the residue field $\bar{\mathbb{F}}_p$ of $K^{ab}$. Both statements are genuinely global. To give you an idea: class field theory of local function fields is very very similar to class field theory of $p$-adic fields. But the analogues of the two statements you are quoting are different for number fields: the kernel is bigger, whereas the image is actually all of $Gal(K^{ab}/K)$.