This is pretty interesting. Assuming a rotation based equilibrium, I suppose the quick rotation could explain the short day cycles.
Though, you have to consider that the inner and outer equators are very different. Depending on the rotational axis and axial tilt, the inner would sometimes be blocked by the opposite half of the torus, like a half-eclipse or a second night. If the axis of rotation was not parallel to the axis of rotation (which isn’t a problem you see on a spherically symmetrical planet like Earth because you could just change where the rotational poles are) this would decrease the frequency of these self-eclipses, and would undermine the concept of seasons. Alternatively, if they were perpendicular but the planet had a low orbit around a dwarf star, with a decent axial tilt, the seasons would be a lot faster, and would receive sun cycles for months, but the inner equator would experience some very heavy eclipsial winters. For a system like Greg described with an internal spinning magnetic ring, you could just have it spin perpendicular to its revolutionary axis, so the inner equator receives slightly less light on average but seasons can still exist. Regardless of which of these systems are used, it would be more complex than earth, including on which points are “tropic” “temperate” and “frigid zones”.
As for the distortion of the planet, I’ve tried to do some calculations. Assuming the z-axis is the revolutionary axis, here is the function of the x, y, and z components of the gravitational field of a single point mass on a torus, where x, y, and z are the position of the ‘test mass’, and theta and r are the polar coordinates of the point mass.
If I do a definite integral of this d theta from 0 to 2pi, this would give me the force field of the entire ‘hoop mass’ torus, but the equation is, as far as I can tell, not integrable, so I did a 32-step left Riemann sum (and the resulting equation is too long to write here. The equation would be even more complicated if I did a fully 3d one with thickness and everything. Meanwhile, the magnitude of the centripetal acceleration is equal to the angular frequency squared over r. The angular frequency (2pi over the day length) in a rotation based torus should be (where g is gravitational acceleration and ac is centripetal acceleration) such that g+ac = 0 when the radius of the test point is equal to the radius of the torus’s ‘mass hoop’, which is surprisingly hard to calculate on my approximation. Back on track to the distortion of the shape, the ‘equilibrium shape’ which a planet will distort to is approximated by the orthogonal trajectories of the force field, assuming the center/hoop of mass never changes. Using pplane8 for matlab, I calculated the distortion for a non-rotating torus, a rotating equilibrium torus, and earth, respectively:
To make sense of these images, this represents a cross-section of the planet. For the tori, it’s actually the right half of a cross sections, where y=0 (the x axis, disappointingly) is the revolutionary axis. When the lines go off the page, that means that equilibrium point is unstable, or in other words, the centripetal force is higher than the gravitational force, so at conservative points, dust, rocks, horses, mountains, etc. are not restrained to the ground and may at any moment rapture themselves to a planetary escape velocity, and mass slowly escapes the planet, giving the planet a ring, or that is to say, another ring, which probably won’t last very long. And at more concerning points, the planet is torn apart in seconds. Non-rotational torus planets tend to form an egg-shaped, with the pointy bit toward the center of the torus. Rotational systems do the same thing in the opposite direction, but have a maximum size. And a fast spinning Earth would be surprisingly American-football shaped planet.
I’m not sure how the wind would work on a rotational torus planet, it’s complicated as hell, your diagram may be a good enough estimate though. And the extended Coriolis effect may have some major impacts on the distribution of the tectonic plates.
Biodiversity between the islands would certainly be a lot greater than on earth, but some animals like birds, seals, flying insects, etc. migrate incredibly far, and on rare occasions can bring seeds, bacteria, fungi with them, which may become invasive species and become prominent on more than one island. Aquatic reptiles like turtles may also re-evolve to return to land on a separate island, and it’s less likely but even mammaliforms like whales could as well. So having each island be the only place you find that class of animal is unlikely, but they would still be much more biodiverse than earth.
edit: centrifugal pseudoforce, not centripetal