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".
- 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 $H=diag(K)$.
- 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
- $s_xx=x$
- $s_xs_xy=y$
- $s_x(s_yz)=s_{s_xy}(s_xz)$
- $x$ 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 $s_xy=2x-y$.
- 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).
- 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]. https://www.jstage.jst.go.jp/browse/tmj1911
