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Moreover, for Note 2, if the solid is compact and its surface is $C^2$, then, for any point $p$ on the surface, there will always a point $q$ for which 'the' shortest geodesic joining $p$ to $q$ is not unique. You should look up 'cut locus' on Wikipedia to see what I mean.
So now that you've clarified Note 1, I can say that, if the surface is $C^2$, then no surface (compact or not) has the property that every pair of points can be joined by two distinct geodesics of shortest length. In fact, every point $p$ of such a surface $S$ has an open neighborhood $U$ such that any two points in $U$ have a unique shortest geodesic in $S$ joining them. (Of course, I don't count reparametrizations of a geodesic as different geodesics.).
I don't understand your Note 1. There are more than 4 geodesics joining the centers of opposite faces of a cube. It's true that there are only 4 that minimize length, but you haven't (so far) required geodesics to have the minimum length of a curve joining the two points.
I think you need to ask a more precise version of your 'second query'. The parenthetical remark at the end seems to be asking about a different question.
It can happen, though, that there are no almost complex submanifolds of $M$ of dimension higher than 2 until you get to $M$ itself. For example, the $\mathrm{G}_2$-invariant almost complex structure on $S^6$ has no almost complex submanifolds of dimension $4$.
Actually, any almost complex manifold $M$ has lots of almost complex submanifolds of real dimension $2$, and the almost complex structure on $M$ restricts to all of these 'pseudoholmorphic curves' to be integrable.
If a surface is smooth (or merely $C^2$), having all its geodesics be planar is equivalent to the surface consisting entirely of umbilic points, i.e., the surface has to be part of a plane or a sphere. This is a simple consequence of the structure equations.
What do you mean by "necessarily a disk"? There are well-known examples of Möbius strips with Gauss curvature equal to zero. One can easily make one with a strip of paper.
@AntonPetrunin: I think that there is, especially since you are assuming a compact, flat minimal submanifold. In my TAMS paper, I assumed only that there was a small piece of a flat minimal surface in the $n$-sphere and the result was that each small piece extends to a complete minimal immersion, and the condition that it close up to a compact surface is even more restrictive. In the global case, it's already clear that, for a given flat $n$-torus $T$, the space of minimal isometric embeddings into any $S^N$ (up to congruence) is finite dimensional.
Since $\exp:S_d(\mathbb{R})\to S^+_d(\mathbb{R}$) is a diffeomorphism, where $S_d(\mathbb{R})$ is the vector space of symmetric $d$-by-$d$ matrices with real entries (whose dimension is $\tfrac12d(d{+}1)$), and $S^+_d(\mathbb{R})\subset S_d(\mathbb{R})$ is the open cone of positive definite symmetric $d$-by-$d$ matrices, any injection $f:\mathbb{R}^d\to S^+_d(\mathbb{R})$ is uniquely of the form $f = \exp\circ\phi$, where $\phi:\mathbb{R}^d\to S_d(\mathbb{R})$ is an injective map, so you are reduced to 'characterizing' the injective maps $\phi$ into the vector space $S_d(\mathbb{R})$.
