What determines the half-life (or stability) of an isotope? Why is it that some elements do not have any stable isotopes, while others of very similar mass have multiple stable isotopes? What determines this phenomenon and what determines the strict periodicity of an unstable isotope's half-life?
 A: Here is a somewhat simplified explanation.
The nucleus of an atom is populated with protons and neutrons, which behave as if they were tiny little spheres with well-defined radii. The strong force holds them all together, and in this state they are all essentially touching their nearest neighbors inside the nucleus.
Like the electron shells surrounding them, nuclei too have quantized energy levels inside them and the protons and neutrons occupy those levels. Since those nucleons resist being deformed, and since it is not possible in all cases to stack them all together into perfectly spherical shells, the resulting nuclei are not perfectly spherical either- and the amount of energy stored in those imperfect packing schemes varies from one type of nucleus to another.
Finally, consider the fact that when bound closely together by the strong force, two adjacent protons are actually repelling each other very strongly- so strongly, in fact, that neutrons must be included in the nuclear mix to help glue those protons together. But since the nucleon packing is imperfect, so also is the distribution of attractions and repulsions between nearest neighbors. This in turn means that some nuclei are naturally less stable to perturbations and upsets than others- and some of them would for example very much like to be rid of a proton or two. But getting rid of a couple of protons is easiest if you also get rid of a couple of neutrons at the same time, since two protons and two neutrons in one glob (an alpha particle) is an exceptionally stable configuration.
To spit out an alpha particle in the interests of reducing the energy of the nucleus and make it more stable requires a certain amount of activation energy, which over time will be furnished by random chance. The lower the activation energy, the more likely in any given slice of time it will be that the alpha gets ejected, which yields a short half-life. The higher the activation energy, the less likely, and you have a longer half-life.
