Well the object itself doesn’t really become 2-d, it just appears that way to outside observers.
Right, which is one of my fundamental disconnects (read: problems) with how objects at speed are discussed.
How it appears to an outside observer is completely inconsequential. The object itself is still the same size/shape it was at rest because the entirety of the object is moving at the given speed.
Which leads to another part of the discussion - the question over whether turning on a flash light at light speed would project light forward onto something likewise moving at the speed of light. Again without being one of the aforementioned mathematicians I say “yes, it does” because I can’t see why the flash light/filament/photons care that they’re already moving at the speed of light. The source and target are comparatively stationary.
If the target were moving away from the source at the “speed of light”, sure, the light never makes it to the target because “that’s how fast light goes”.
The real question should be - “why is the speed of light...the speed of light?” There is so much we don’t know about this universe, and likely never will.