If you call it "reciprocity failure", people are more likely to know what you mean. Yes, your description is about right: you have photons arriving randomly (a Poisson process, to be precise) at each sensitive crystal. A partially-sensitised crystal will return to its base state by leaking the captured electron - also a random process but I think with a narrow distribution, so you can think of it as taking a fixed amount of time to lose the electron.
There is no absolute minimum level because as you noted, photon arrival (from a natural scene, not a controlled source as used in a two-slit experiment) is random. Say you need two photons to form a stable latent image, there is no minimum level of illumination at which you can't by luck get a pair arriving sufficiently close together. Of course, the likelihood drops off rapidly with reducing illumination. Yes, you can make a year-long exposure and there are several decent examples on the web showing building construction, paths taken by the sun throughout the year, etc - typical approach is to use low-sensitivity medium like paper (also has reciprocity failure) and a pinhole.
Films like Acros have excellent reciprocity characteristics because they (mostly) need only a single photon to reach a stable exposed configuration.
Films are sensitive only to photons above a certain energy (proportional to their frequency); longer-wave (redder/colder) photons will not be absorbed by the film. So most thermal photons have no effect. This is the reason why IR films are sensitised no further than 1um - if they were, they would have to be kept refrigerated their whole life and even used in a refrigerated camera with a refrigerated shutter & refrigerated lens. And developed at low temperature, which is even harder.