The moving Sensor is effectively for image stabilization. Sony is the only company that has that stabilization in the body of the camera. Other camera manufactures have the stabilization in the lens.
Which brings up an interesting point. The cost of a lens for a Sony is less than the cost of the equivalent lenses from any other manufacturer. Every lens for Canon, Nikon, etc, cost more to manufacture, because of the stabilization that has to be in each Lens.
But on the other hand, it makes the body heavier, and more susceptible to being damaged in a drop. Big deal you say, you don't go around dropping expensive Camera bodies, right? Well, Wedding photographers drop their camera bodies all the time. Hence the reason they always have at least two, sometimes three camera bodies at their disposal. So the reasoning for a Camera body without stabilization is that it is faster, easier and most importantly cheaper to replace a Camera body without all the bells and whistles of a Sony. Dropping a lens at a wedding can be expensive but you still have the body to use with other lenses. Drop the body and you are SOL.
But take it all this with a grain of Salt. As in every other 'this vs, that' discussion (Apple vs. PC, Ford vs. Dodge) there are heated discussions about who or what is better.
And yes, you are correct in saying that 35mm Film is the digital equivalent to Full Frame. And there again, there are different schools of thought as to which of the two most common formats is better (APS-C is the other one.) The same goes for internal/built in flash vs. external flash.
Now I know what is inside my DSLR.
A braver man than me!
Presumably the moving sensor allows for image stabilization in the presence of camera vibration or shake.