(1) Is it correct then that geodesics in M (if M is Riemannian) are just a special case of this construction?
Yes, it is correct to say this. Give a Riemannian manifold $M$ with a metric $g_{ab}$ and an affine connection $\Gamma^a_{bc}$, one can construct various geometric invariants on $M$ such as the Riemann and Ricci tensors. Now for a point particle propagating on this static manifold, i.e without taking General Relativity into account, one could write various Lagrangians of the form:
$$ L = \sqrt{|g|} ( K_m - V_m) $$
where $|g|$ is the absolute value of the determinant of $g_{ab}$ and $K_m$ and $V_m$ are kinetic and potential terms respectively which depending on the position and velocity of the particle or more generically is energy-momentum n-vector $j^a$. A generic example of the kinetic term in this case is:
$$ K_m := \frac{1}{2} \left( \partial^a j_a \right)^2 + \frac{1}{2}(\partial^a j^b \partial_a j_b) $$
and $V_m$ is some polynomial in $j^a$.
If we wish to incorporate the dynamics of the manifold itself (i.e. move into the domain of GR) then our Lagrangians must contain "kinetic" and "potential" terms for the various geometric invariants. These quantities however lack a simple expression for a general manifold and are subsumed into the Ricci scalar:
$$ L = \sqrt{|g|}(K_m - V_m + \mathcal{R}) $$
where the Ricci scalar $\mathcal{R} = g^{ab}\mathcal{R}_{ab} = g^{ab}g^{cd} \mathcal{R}_{acdb} $ is the trace of the Ricci tensor which in turn is a partial trace of the Riemann tensor.
Now, geodesics on a Riemannian manifold $M$ with a metric $g_{ab}$ and an affine connection $\Gamma^a_{bc}$ are given by solutions (integral curves) of the of the geodesic equation:
$$ v^a\nabla_a v^b = 0 $$
where $v := v^a \partial/\partial_a $ is a vector field on $M$, the covariant derivative is given in terms of the affine connection $\nabla_a v_b = \partial_a v_b + \Gamma^c_{ab}v_c $ and indices are raised and lowered with the metric $ v^a = g^{ab} v_b $. Your question amount to asking if there exists a Lagrangian $L(v^a)$ whose equation of motion is this geodesic equation. The answer is 'yes'. The following Lagrangian
$$ L = \sqrt{g^{ab} v_a v_b} $$
which is nothing more than the proper distance along the particle's wordline, yields the geodesic equation on variation w.r.t $\delta v_a$.
(2) Homogeneous spaces, i.e. manifolds of the form G/K, where G is a Lie group and K a closed subgroup, provide interesting examples of manifolds. What would a physical system look like which has G/K as a configuration space? I remember hearing something along the lines "global G-invariance, local K-invariance" but I'm not sure.
An example of a physical system with this configuration space is MacDowell-Mansouri gravity. A beautifully written reference for this is the review paper MacDowell-Mansouri gravity and Cartan geometry by Derek Wise, a former student of John Baez.
There one starts with a five-dimensional spacetime with symmetry group $G$ which can be deSitter, Anti-deSitter or Minkowski with a lie-algebra valued antisymmetric tensor $B$ and the curvature of the gauge connection $F$. The action is that of a topological BF theory given by:
$$ S_{BF} = \int tr B \wedge F $$
We can identify $B:= B_{\mu\nu}^{IJ}$ with the wedge product of two fermion fields: $ B^{IJ} = \psi^I \wedge \psi^J $ which transform in the fundamental representation of the gauge group. Furthermore, as in the case of the BCS mechanism, let us assume that the dynamics of the system contains a four-fermion term which acquires a vev (vacuum expectation value) due to the formation of a condensate of these fermions. Such a term results in the action:
$$ S'_{BF} = \int tr B \wedge F - \frac{G\Lambda}{6} B \wedge \star B $$
where $\star B$ is the Hodge dual of $B$. $G$ and $\Lambda$ are Newton's constant and the cosmological constant respectively. The formation of the condensate can be physically described by writing the five-dimensional gauge connection in the following form:
$$ {}^5 A =
\left( \begin{array}{cc}
{}^4 A && \frac{1}{l}\{e^0,e^{i} \} \\
\frac{1}{l} \{ e^0, \epsilon e^i \} && 0
\end{array} \right) $$
where ${}^4 A$ is a four-dimensional connection and $\{e^0,e^{i}\}$ (where $ i \in {1,2,3}$) is a vier-bien (tetrad). $\epsilon = \{-1,0,1\}$ for $G = \{ SO(4,1)$ (deSitter), $ISO(3,1)$ (Minkowski), $SO(3,2)$ (Anti-deSitter) $\}$ respectively. The group $H$ in each case is $SO(3,1)$ and the resulting theory has gauge group $\{SO(4,1)/SO(3,1), ISO(3,1)/SO(3,1),SO(3,2)/SO(3,1)\}$ respectively.
The resulting theory describes general relativity in four dimensions with the addition of topological terms such as the Nieh-Yan and Pontyargin terms in the action. For more details see Wise's excellent paper reference above and also the seminal paper (1) by Friedel and Starodubstev who first proposed this formulation of the MacDowell-Mansouri mechanism.
