Why does it took so long to cool down? I am by no means an expert on nuclear reactors. If the rods went into a shut down position, why is there still heating of the reactor core? Please be patient with me on this one.
Well simply said, the fuel elements of a nuclear reactor are similar to the plutonium fuel elements inside the RTG of a spacecraft. In a RTG, a slowly decaying substance like Plutonium produces heat energy by nuclear decay. Every second, a fraction of the available fuel decays. The energy behind every decay is pretty much constant for the type of decay, the number of decays per second and the involved isotopes defines how thermal power the decay of a kg of fuel has.
In a nuclear reactor, after shutdown, there are many isotopes in the fuel elements and the water, that decay quickly, producing much more energy per kg for a shorter time. Also, there is a lot more material in the reactor, than in a RTG. Thus the need for cooling even after the control rods reduced the neutron flux a lot.
The neutron flux drops pretty rapidly after a scram, especially because of Xenon poisoning. If the water boils away, there is also less moderation and less thermal neutrons (neutrons that are slow enough for triggering nuclear fission) are around, which means the risk of the core getting critical is even lower when the water boils (As you can learn in the simulator "BWR", hotter water means less reactivity).
BUT: Steam has only a low thermal conductivity. Especially dry steam, that is steam that is hotter than just the boiling point. The steam reduces the heat by nuclear decay in the fuel elements less good as water. The temperatures can easily raise. Next, once the surface of the fuel elements exceeds 800°C, the zircalloy cladding of the fuel (that has the purpose of keeping the fuel in the fuel rods) reacts with the steam and this chemical reaction produces a lot of hydrogen and more heat.
Also, what can make such situations even more bad: Lets assume the normal cooling fails. In this case, you need to use alternate cooling systems. There is a high pressure one, that you can use at operating pressure of the core, and a low pressure one, that requires you to vent steam first and reduce the pressure in the core. If you have a massive circulation pump failure, the flow from the high pressure system can be not enough to keep the water level in the reactor above the fuel. So, the usual action is to automatically vent the pressure into the suppression pool. But reducing pressure also means that the boiling point of water drops and more steam is produced. Which means you need to pump much more water into the reactor at a time to compensate.
This essentially happened in Japan. First the normal shutdown by automatic primary seismic wave triggered scram happened, all pumps worked. temperatures had been slowly cooling.
Then the Tsunami washed away the pumps and/or the fuel tanks for the Diesel generators. Now most of the pumps failed, because electricity was gone. Only thing left had been a few ECCS pumps that had been able to operate on battery power. Temperatures and pressure starts to raise. These pumps had not been designed for operating for a long time.
So, for buying time, the pressure was reduced, more flow was possible by using the low pressure spray system, but at the same time, more water rapidly boiled away.
The water levels in the reactor drop, despite more water flowing into it. At the same time, the temperatures in the suppression pool reach boiling point as well, and the water levels there drop as well. Less water is available for cooling by the only available water source.
When the final battery pack is consumed, there is no flow at all anymore. Only weak convection inside the too small reactor vessel provides some motion and heat transfer.
While the water levels drop, more fuel gets exposed and heats faster. Now the fuel cladding starts to react with the steam, and hydrogen forms. The hydrogen is vented with the steam into the suppression pool.
When the pressure in the suppression pool becomes too high, air from the containment building is vented into the reactor building (Not directly into the atmosphere) The hydrogen concentrates below the roof, until there is an explosion.
By pumping cold sea water into the suppression pool, there is already new water available for venting steam and pumping more water by low pressure into the core.
At least one reactor had also the whole containment structure being flooded (including the pumps) as last resort action, to provide more cooling (bigger heat transfer from inside to the bigger outside area). This likely damaged the pumps inside as well. Such pumps are robust and should work even when water sprays on them or steam condenses at them, but I am not aware that being 100% water tight was ever required.
