Are there any physicists (or other relativists) who will tackle some relativity paradoxes?

We are told that according to special relativity, SR, anything moving at the speed of light should have a zero time rate and zero length in the direction of relative motion. This is because of the Lorentz factor which is infinite at the speed of light. So it seems light should have no observed frequency or wavelength!

If instead the factor of infinity is applied to the measured properties of light then in its own reference frame it would have infinite frequency and wavelength. But an infinite wavelength implies a frequency of zero not infinity (the product of frequency and wavelength being the constant c).

It seems obvious that light behaves quite differently to matter and SR does not apply to light. Yet photons are said to have zero rest mass. They only have mass and energy because their zero rest mass is multiplied by infinity using SR. Hence if SR doesn’t apply to light it would have no mass, energy, frequency or even existence. So does SR apply to light or not?