The cutting is occurring under water; I'm sure that the Japanese, depending on their level of desperation, can build a test scenario to replicate the conditions in the reactor, and make adjustments before sending anything in.
Again, the core itself is under water...you might need to move some debris to get around things, at which point multiple robots, or perhaps, multiple robots with winches / more powerful / disposable tools might be useful. If it requires two of them to remove a girder to get to the sunken core, then two of them they will need. I favor the laser because it's bound to be useful for cutting the melted rods into chunks you can safely remove...mechanical attempts would be iffy, at best. And it might be possible to keep the actual laser generation apparatus outside the reactor, while feeding the cutting beam through the fiber optic cables. This would allow complete control of the laser + shutdown of the laser in an emergency + keep it out of any danger / radiation. Plus chemical lasers tend to be stronger than other kinds...or that was the impression I was left with last time I checked...things may have changed.
And yes, you are right about using the laser itself on the cores. Some double-checking would be needed to ensure that it would not 'assist' any additional neutron release, but I'm sure the idea itself is still a sound one.
As the core is dissected, I'd direct the robots to place each piece in a lead-lined storage pod; this needs to be done as each piece is cut off, so as to not create further metldowns; each piece must be placed in its own self-contained pod. After this is done, the immediate worry about needing to constantly pump in coolant to keep the core from getting worse will be alleviated...and the core itself will be contained, and presumably taken somewhere to be recycled or stored or buried. Continuing to pump water into the reactor, when it already contains seawater (WTF), which is a serious no-no for typical nuclear chemistry, is just a recipe for disaster. Your choices are many, but the ones here are 1.) keep pumping in water and watch it keep poisoning things (which does seem to be the media's current assessment...no idea whether that is true), or 2.) dismantle the core so that radiation will drop. There is, of course, as you highlighted, the possibly of dust creation...but lasers tend to be pretty good here; your choices are 200 years of highly radioactive water flowing out of the reactor (need to verify this, is it actually highly radioactive and flowing? and 200 years worth?) or what will probably be a temporary spike of whatever dust is created by the lasers used (better than other cutting tools, I imagine), followed by an immediate drop off as the core is sealed. And the dust created will be somewhat granular, so perhaps some careful planning there can limit its spread as well (I suppose if you want to get extra crazy, you would apply an airlock or something to the area directly above wherever you find the melted core...and just filter for particles larger than 0.3 microns or something and smaller than the robot. You could apply gel to the cut pieces of the cores itself, to prevent any dust from moving, thought that would happen afterwards.