Nothing can be infinity FPS because it would be limited by the frequency of the light that we use to see the watch hand. In the case of the visual spectrum, it caps out around 668 THz with blue light (red light is lowest at 400 THz). This would make it "only" ~4.7E12 times better than 144 Hz ;-).
With that said, though, I believe that the movement of the actual atoms making up the mechanical hands is still happening in an "infinitely smooth" sense, although the way we measure and observe that movement (via reflections of light or perhaps some other type of electromagnetic radiation) may be limited.
>There is currently no proven physical significance of the Planck length;...
It's just dimensional analytical playing with some fundamental physical constants. Physicist believe that this scale length may have some connection with the yet to be discovered laws of quantum gravity. It has nothing to with the "finite resolution of the universe". There is no evidence that the universe has finite resolution.