First, my apologies for this late answer, I only found the question today. Below, I probably recall too many things, but I felt it could put some context around the short answer to question 1 saying: yes, under the names "geodesic in an involutory quandle", or "cycle in a symmetric set".

1. Recall that a homogeneous symmetric space (I borrow the terminology from Loos [2]) is a homogeneous space $G/H$ of a Lie group $G$, where $H$ is an open subgroup of the fixed points subgroup $G^\sigma$ of an involution $\sigma$ of $G$ ($\sigma\in Aut(G)$ and $\sigma^2=Id$).

A homogeneous symmetric space has a canonical connection $\nabla$ for which the geodesics are of the kind $t\mapsto g\cdot \exp(tX) H$ with $g\in G$ and $X\in \mathfrak{p}$. Here $\mathfrak p=\{X\in Lie(G) \mid Lie(\sigma)(X)=-X\}$.

Every Lie group $K$ (compact or not) is a homogeneous symmetric space, with $G=K\times K$, $\sigma(k,k')=(k',k)$ and $H=diag(K)$.

2. There is an intrinsic presentation of symmetric spaces due to Loos: a symmetric space is a smooth manifold with a smooth product law $(x,y)\mapsto x\bullet y=s_xy$ such that

1. $s_xx=x$
2. $s_xs_xy=y$
3. $s_x(s_yz)=s_{s_xy}(s_xz)$
4. $x$ is an isolated fixed point of $s_x$

The maps $s_x:M\to M$ are called the symmetries. A morphism of symmetric spaces $M\to N$ is a smooth map $\phi:M\to N$ such that $\phi(s_xy)=s_{\phi(x)}\phi(y)$. Any homogeneous symmetric space $G/H$ is a symmetric space in this sense, with $s_{gH}(g'H) = g\sigma(g^{-1}g')H$. Conversely, any connected symmetric space $M$ (with a choice of base point $o\in M$) is a homogeneous symmetric space $G/H$, with $G$ the subgroup of $Aut(M)$ generated by $\{s_xs_y\mid x,y\in M\}$ (it is a finite-dimensional Lie group), involution $\sigma(g)=s_ogs_o$, and $H=Stab_G(o)$.

In this context, it can be seen that a geodesic in $M$ is simply a morphism of symmetric spaces from the real line $\mathbb R$ to $M$. Here, $\mathbb R$ has the symmetries $s_xy=2x-y$.

3. If we remove axiom 4. in the definition of symmetric space (and forget about smoothness), we get a purely algebraic object which appears under various names in the literature: kei (Takasaki [5]), symmetric set (Nobusawa [3], for finite sets), involutory quandle (Joyce [1]), symmetric groupoid (Pierce [4]).

In analogy with the smooth case, one may define a geodesic in $M$ as a morphism of such spaces, from the integers $\mathbb Z$ to $M$. Here, $\mathbb Z$ has the symmetries $s_xy=2x-y$.

Joyce gives an abstract definition of involutory quandle with geodesics [1, p. 30] and shows that any involutory quandle can be seen as an involutory quandle with geodesics, essentially by defining the geodesics as is done above.

Obviously, a geodesic is determined by the images of 0 and 1 (the point-and-tangent-vector datum determining a geodesic in the smooth case is replaced by a pair of points in the discrete case). Nobusawa [3, pp. 570-571] calls these geodesics cycles (symmetric subspaces generated by two points).

As in the smooth case, any (finite) group $G$ can be seen as a (finite) symmetric set by setting $s_gh=gh^{-1}g$. In the finite group case, the geodesics then coincide with your definition (except that singletons are not excluded).

4. That said, to my knowledge, these geodesics have not been much studied for themselves. I don't know the answer to questions 2 and 3 (except that in the present context, no metric is involved).

References

[1] David Joyce, An Algebraic Approach to Symmetry with Applications to Knot Theory, Thesis. http://aleph0.clarku.edu/~djoyce/quandles/aaatswatkt.pdf

[2] Ottmar Loos, Symmetric Spaces. 1: General theory, Benjamin, New York, Amsterdam, 1969.

[3] Nobuo Nobusawa, On symmetric structure of a finite set, Osaka J. Math. Volume 11, Number 3 (1974), 569-575. http://projecteuclid.org/euclid.ojm/1200757525

[4] R. S. Pierce, Symmetric groupoids, Osaka J. Math. Volume 15, Number 1 (1978), 51-76. http://projecteuclid.org/euclid.ojm/1200770903

[5] M. Takasaki, Abstractions of symmetric functions, Tohoku Math. J. 49 (1943), 143-207, [Japanese].