Hi Tom, I might know some of the answers as I have been interested in nuclear energy. But if anyone knows better correct me if I am wrong.
The water that escaped from the nuclear stations in Japan may have leaked out of the cooling system, it is possible the water could have escaped from the turbine side of the power plant. The GE boiling water reactor is popular in Japan and that has slightly radioactive water going to the turbines. Hopefully, it wasn't much and won't do much damage. It is likely the reactor "scrammed" itself and shut down. But the utility probably wants to get some of the nuclear power plants going again. Fossil fueled plants are probably damaged and electricity is needed to assist in the recovery efforts.
The barrels that fell over likely had "low level nuclear waste", items such as filters, gloves, protective clothing, boots, wipes, things that are used to maintain the facility. This items should be dry and can be placed back in the barrels. They probably wait until they have enough barrels for a truckload, then take them to a waste facility, whatever they may have. Some lab tests in hospitals use small amounts of radioactive material and this is how they handle such waste.
High level waste is the spent fuel rods that have been unloaded from the reactor after being used for a year or more. In France, for example, this waste, which would be solid, is melted into the glass (vitrification) where it would be unlikely to be released into the environment. In the US, such waste will eventually be taken to Yucca Mountain in Nevada. Right now, spent fuel rods are keep in extra-deep "swimming pools" at nuclear power plants to "cool off" and lose some of their intense radiation.
Reprocessing nuclear waste is possible and is practiced by Areva in the Le Hague plant in the north of France. Spent fuel rods, in a sealed chamber, are dissolved in nitric acid and the elements such as the unused uranium and plutonium are filtered out and can be used to fabricate new fuel. Fuel with plutonium is "mixed oxide fuel (MOX)" and can be used in most existing reactors. Becuase of worries of nuclear proliferation (plutonium is the stuff of bombs), President Carter banned reprocessing in the US in 1977 and no President since has been interested enough to even bring up the issue. A lot of nuclear waste has also been created by nuclear weapons programs, this will also go to Yucca Mountain (if Congress approves).
I encourage you to go to the website of the Nuclear Energy Institute - it is pro nuclear but it is very informative.