With lunar Thorium being common, and heavier than iron or nickel, does Earth's core have the occasional nuclear reaction?
The idea of a natural nuclear reactor exists, and in fact there is a claim of one such site in Africa where enhanced uranium fission is suspected to have occurred a couple billion years ago. It was not thorium, however, and in fact the linked article claims this is the only such natural reactor found to date. It's harder to directly observe things deep inside the Earth, but as dmckee points out, any large enough reactor at the Earth's core would have been detected from its neutrino signature.
As for "the occasional nuclear reaction," this happens all the time. Reactors are merely mechanisms that enhance the rate of nuclear reactions to create useful power levels from small amounts of fuel. But any unstable isotope will by definition decay over time. In fact, part of the reason the Earth's core is so hot is that it is kept warm by all the isolated nuclear decays occurring.