A very peculiar fact is that in a compact space THERE IS a preferred inertial system. Indeed even if locally there is no way to single out a preferred inertial system, globally you can do it.
Is the topology that tells you that an observer doing a loop around a torus is topologically different from an observer moving around simply connected loops.
So for instance, the twin paradox is solved in this space too, and the observer traveling around a not simply connected loop is younger than his twin. For references see: Twin Paradox in a Torus
For the second question, I guess that that dimension would be negligible from the point of view of extended dimensions. Nevertheless many quantum gravity theories seems to suggest that spacetime is an emergent phenomenon so I'm not sure that insisting with the classical spacetime picture at energies near the Planck scale is still meaningful.