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1. The fuel will continue to make heat after shutdown due to decay heat from radioactive decay. However after that decay the material that decays will no longer be radioactive, and thus no longer produce heat. The rate of decay depends on the element that is radioactive, this is called a half life. That is the amount of time it requires for half of the material to emit radiation. During operation 7% of the heat from the reactor comes from decay heat. After the reactor shuts down due to a scram (full insertion of the control rods), 3% of the previous power will still be generated due to decay heat. After about an hour that will drop to 1.5% as the radionuclides with a half life of less than ten minutes eliminate themselves. The only way to shut off the heat is time, or by exposing it to a neutron flux to physically change the fission products into other things (transmutation).
2. There are multiple levels of containment, the first stage is the rod itself, so long as the cladding on the fuel assembly is not damaged, fission products will not leak from the fuel. The next step is the pressure vessel, this is how they circulate water around the rods to remove and harvest the heat for later use in a steam turbine. This system has penetrations that are specifically designed to allow for the discharge of fluids. The discharge system is what was responsible for the hydrogen explosion, it penetrates the third containment structure and vents to tanks located inside the turbine building. The third containment in a boiling water reactor is called the drywell. It's basically designed as a worst case failsafe in case the pressure vessel melts to catch the resultant molten slag from the core. When they began pumping seawater it's likely that they were actually filling the drywell itself up with seawater in an attempt to increase heat lost to ambient from the reactor. Air is not as good as water at conducting heat, so if they surround the pressure vessel with water instead of air it will be more easily cooled.
3. The rods in the spent fuel rod pool continue to produce heat, but after a while it's possible to remove them to dry cask storage, this is being done all of the US today. I don't remember the exact time frame, but considering the plants currently operating are all under 60 years old, it takes less time than that. I believe it's somewhere from 5 to 30 years, but I'd need to do more research.
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