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You definition allows that (for three points A,B,C) the curves from A and B to C come to C with the same velocity vector which is not possible for geodesics. … definition does not require that the segment of $\gamma_{ABC}$ from $B$ to $C$ is a trajectory of your exponential map starting from $B$ and going to $C$, though this property may be essential for geodesics …

Elipsoid does not posess unbounded geodesics with no self-intersection. I do not know a conceptual explanation. … The second type are already closed geodesics ( and on the ellipsoid there are at most 3 of such geodesics of the second type). …

I assume that imranal asks how to find numerically a geodesic connecting two given points if the connection is given. One way to do it is to implement the solution of the ODE system he wrot …

This is an answer to the last question of the revised version of your question which is "What if instead of the the parallel almost complex structure we assume that there is a nonzero parallel vecto …

Let me give a standard example of a closed incomplete manifold with flat affine structure, whose 2-dimensional version is essentially the example from the comment of Misha. Consider $R^n\setminus 0$ …