Thinking about the "objects falling in a mineshaft through the earth" scenario, this doesn't work in general terms as coriolis effects would require a figure 8 orbit , (else the objects would scrape the inside of the shaft) requiring a figure 8 mineshaft. However if we expand the size of the "mineshaft" to that of the hole through a torus, a figure 8 orbit would be achievable.
Obviously many figure eight orbits would exist of a moon around a non-rotating donut world (or one spinning about it's axis of revolution) , but what if the donut were spinning end on , (or tilted). There would be some subset of figure eight orbits that would be spin synchronous. i.e. the moon passes through the hole at say 0degrees, then a little while later, the donut has rotated 90degrees and the moon goes back through the hole again. It would seem there are many interesting orbital possibilities here, not sure whether you could call them "precessional" or "coriolis" or "schuler" or "harmonic" orbits .
So the question: how would one calculate these orbits, and how to determine if tentative orbits are stable (i.e. some partial derivative would determine re-convergence as a consequence of minor pertubations (or tidal drag) . .
I have drawn a "time lapse" view , looking along the spin axis , at 4 time intervals 1 , 2 , 3 , 4 . Interestingly enough, a viewer on the "outer" surface of the planet would see the moon shoot across the sky from east to west, then from west to east each day , while a viewer on the inside would see 4 passes per day E-W , W-E, E-W, W-E. Other places on donut world would see the moon stop and retrace it's path.
