A nuclear reactor operates just below critical mass so that each radioactive decay in its fuel rods induces a large but finite number of subsequent fissions. Since each chain reaction gradually weakens away to nothing, there is no danger that the fuel will explode. But operating just below critical mass is a tricky business and it involves careful control of the environment around the nuclear fuel rods. The operators use neutron absorbing control rods to dampen the chain reactions and keep the fuel just below critical mass.

Fortunately, there are several effects that make controlled operation of a reactor relatively easy. Most importantly, some of the neutrons involved in the chain reactions are delayed because they come from radioactive decay processes. These delayed neutrons slow the reactor's response to changes—the chain reactions take time to grow stronger and they take time to grow weaker. As a result, it's possible for a reactor to exceed critical mass briefly without experiencing the exponentially growing chain reactions that we associate with nuclear explosions. In fact, the only nuclear reactor that ever experienced these exponentially growing chain reactions was Chernobyl. That flawed and mishandled reactor went so far into the super-critical regime that even the neutron delaying effects couldn't prevent exponential chain reactions from occurring. The reactor superheated and ripped itself apart.

First, nuclear waste isn't 100% radioactive atoms. Much of it is radioactively contaminated material—normal materials that contain enough radioactive atoms to be considered hazardous. Second, nuclear reactors don't wait for radioactive materials to decay via spontaneous processes, the ones that are responsible for half-lives. Instead, they induce the radioactive decays using chain reactions. In a nuclear fission reactor, the spontaneous decay of one uranium or plutonium nucleus is used to induce decays in other uranium or plutonium nuclei. In this manner, huge fractions of the uranium or plutonium nuclei can be "used up" in only a few years. In fact, in a nuclear fission bomb, many or most of the uranium or plutonium nuclei are consumed in less than a millionth of a second because of these induced fissions. Half-life has almost nothing to do with a fission bomb. It becomes nuclear waste so fast you can't imagine it.

While the radioactive decays from spent nuclear fuel rods continue to produce thermal energy, the amount of energy released each second isn't enough to make it cost effective to use that energy. Since the power output from a spent fuel rod would only be in the watt range, it wouldn't justify the hazardous job of trying to extract that power without encountering the radiation. Furthermore, the laws of thermodynamics make it much harder to use heat from a warm object than heat from a hot object and spent fuel rods would at best be warm objects.

As you noted, this process of sticking together smaller atomic nuclei or nuclear fragments to form larger atomic nuclei is called fusion. Many smaller nuclei release energy when they grow via fusion, so long as the resulting nuclei are no larger than ⁵⁶Fe (the nuclei of a normal iron atom). Above that size, energy is consumed in the process of sticking the nuclei together. So building carbon nuclei would release energy and building gold atoms would require energy. But while it's possible to construct atomic nuclei up to carbon or even gold, it isn't very practical. It's very difficult to bring atomic nuclei close to one another because they are all positively charged and repel one another fiercely. Because the nuclear energy these nuclei release during fusion only emerges at the moment they actually touch, something must push them together for that to occur. The nuclei can be pushed together by (1) nuclear fission reactors, (2) particle accelerators, (3) thermonuclear weapons, (4) giant lasers, or (5) thermal fusion reactors. None of these systems is ready to synthesize large quantities of normal atoms in a cost effective manner (although nuclear fission reactors do produce useful quantities of radioactive isotopes) and none is ready to produce practical energy from fusion processes.

The loss of cooling water was unexpected and was caused by a pump failure. The broken pump was actually part of the power-generating loop, not the reactor core-cooling loop. When everything was working properly, water flowing through a loop that included the reactor core transferred heat to water flowing through the generating plant loop. But when the generating plant loop shut down, the reactor core loop had nowhere to deposit its heat and the water in it boiled. Backup cooling water evidently did not exist, did not work, or was not sufficient to keep the reactor core from over heating. I don't know whether it was poor design or poor maintenance that caused this disaster.