Super Mario Galaxy problem
This problem is very very difficult.  We could simplify it to make it easier, as follows.


We can add the assumption that the angle sum about every vertex of the polytope $P$ is a rational multiple of $\pi$.  This gets rid of most "polytopes" but there are still many interesting possibilities: for example, the platonic solids. 
We can assume that the polytope is not truly three-dimensional, but instead is the "double" of a polygon; this looks a bit like a pillowcase. We can simplify even further and suppose that the polygon has equal and parallel sides; for example a square, as in the game Astroids.


If we make both of these assumptions then there is a large theory.  (Finding an $O(\log(\ell))$ algorithm for the square is a difficult exercise involving the continued fraction expansion of the angle of Mario's path.  To achieve a similar result for the regular octagon is possible but harder.  The solutions for the square and the octagon involve thinking about how to efficiently encode a "cutting sequences for a geodesic on a translation surface".  Most other rational polygons will quickly lead to open problems.)  An initial reference, which includes a further reference to Caroline Series's discussion of the square torus, are these talk slides by Diana Davis.

If we do not assume rationality, but do assume that the polytope is the double of a polygon, then we are discussing the theory of "cutting sequences in irrational billiards".  It seems that essentially nothing is known here; for example see the final sentence of this talk by Corinna Ulcigrai.

If we make neither assumption, well, I can't think of anything in the literature. 

Finally - I'll guess that there is an $O(\log(\ell))$ solution to the Super Mario Galaxy problem for the platonic solids.  This is a good problem for a graduate student getting started in rational billiards.  For example, the case of the dodecahedron "should" follow from Diana Davis's thesis.  (But start with the tetrahedron - that will follow from an analysis of cutting sequences for the hexagonal torus.)