Torus shaped event horizon Is there a solution to GR field equations for a rotating black hole that has a torus shaped event horizon? If so, when a craft flies through the torus, it can pass through a ring singularity while remaining outside of the event horizon. Would it enter a strange region of repulsive gravity and closed timelike curves? How far from the black hole would this region extend?
 A: The proposition cited in the answer by Nickolas Alves applies to “stationary” and “asymptotically flat” spacetimes, i.e. in four dimensional GR there are no stationary  torus-shaped event horizons of asymptotically flat black hole. If we consider highly dynamical situations such as black hole formation or black hole coalescence, then it is possible to observe horizons for which constant time slices have toroidal topology.
For example here is a figure 10 from the paper:
It depicts an instantaneous configuration of event horizon during the merger of two black holes one of which is rapidly rotating while the other is much heavier (so called extreme mass ratio). We can see (in the zoomed in inset) that there is a small and stretched toroidal “hole” in the “pinch” connecting two black holes. The cited paper also provides references to other works where toroidal transient event horizons have been observed and discussed.

… when a craft flies through the torus, it can pass through a ring singularity while remaining outside of the event horizon.

No. First, ring singularity of the Kerr metric is mostly an artifact of analytic solution of GR, realistic black holes are not expected to have such a feature due to instabilities developing around inner horizon. Second, in scenarios where temporary toroidal  event horizons  form they do not contain ring inside them, for example in the figure above, the ring  singularity would be fully inside the gray spheroid, so spacecraft passing through the hole in a torus would not pass through the singularity. Such a spacecraft (if it stays outside event horizon) would not experienced any “strangeness”.
For completeness, let us also mention that if one drops “asymptotic  flatness” as a requirement than it is possible to construct stationary/static “black holes” with horizons of unusual topology even in four dimensions. But such topological black holes are unrealistic, since in such cases event horizon merely inherits the topology from underlying background and away from the black hole spacetime does not approaches to a flat one.
A: There is a result in four-dimensional General Relativity that forbids such situations. Namely, the Hawking's Theorem on the Topology of Black Holes. It is given, e.g., on the classic book by Hawking and Ellis, The Large Scale Structure of Space-Time, which I quote:

Proposition 9.3.2
Each connected component in $J^+(\mathscr{I}^{-},\overline{\mathscr{M}})$ of the horizon $\partial \mathscr{B}(\tau)$ in a stationary regular predictable space is homeomorphic to a two-sphere.

Hence, at least under this proposition's conditions, you can't have a toroidal black hole. Notice that we are assuming the black hole is stationary, for example, and, implicitly, that we are working with four-dimensional GR. The dimension of spacetime is extremely relevant: in five-dimensions, you do get black rings.
If you're interested, I believe the original reference due to Hawking is Comm. Math. Phys. 25, 152–166 (1972). DOI: 10.1007/BF01877517.
