(i) If $[x,y]$ can be empty, then taking an $n\times n$ square in the square grid and the vertical and horizontal paths as the set of paths $\mathcal C$, all properties are satisfied except the second property for pairs $x,y$ on the boundary; to get all properties satisfied, instead of a square take the vertices of the grid in the lozenge $|x|+|y|\le n$.

(ii) If you relax the third property by allowing that $l(x_1,y_1)$ and $l(x_2,y_2)$ either are adjacent or coincide'', then the 5-cycle with the set of shortest paths as $\mathcal C$ seems to satisfy all conditions.

(iii) Condition (i) needs to be written in a more precise way: as I understand, $[x,y]$ is the (vertex-set) union of the portions between $x$ and $y$ of all paths of $\mathcal C$ passing via $x$ and $y$ (and not their intersection).

(iv) Bibliography remarks: on a related topic (but not for graphs), see the paper R. Dhandapani, J. E. Goodman, A. Holmsen, R. Pollack, S. Smorodinsky, Convexity in Topological Affine Planes. Discrete & Computational Geometry 38(2): 243-257 (2007). About abstract convexity, see the book Theory of convex structures by M. Van de Vel.