It was already tested with supernova 1987A. Neutrinos from that event moved at speed c to within experimental uncertainty. Had they moved at the speed the Opera results indicate, they would have arrived months or years earlier.
No - you'll derive a different set of transformations that are identical to the Lorentz transforms except that the speed involved is different. For instance, you can go through Einstein's 1905 paper and replace "speed of light in vacuum" everywhere with "speed of sound in air at standard temperature and pressure". You'll define a set of coordinate and transformations that look like Lorentz transforms, but with c replaced by 330 m/s or so. In those coordinates, the speed of sound is the same in all "inertial" reference frames (including those that are moving with respect to the air).
However, the physical content of relativity is the assertion that the laws of physics are invariant under Lorentz transformations. That actually guarantees that light must travel at speed c (the speed in the transform), because otherwise Maxwell's equations would not be invariant (they are not invariant under sound "Lorentz" transforms, for instance).
So in short, the speed that appears in Lorentz transforms is a physical quantity that can be measured, has been measured, and the uncertainty in it is far below the level the OPERA collaboration found.