Length contraction and time dilation are words that describe changes that are the consequence of a Lorentz transformation.
Like I have said, the formulae of a Lorentz transformation are defined only when the relative velocity between the two systems is less than the speed of light.
Attempting to pass to a limit when the relative speed approaches the speed of light does not produce any useful result, because at the limit you no longer obtain a reference system, so you no longer get a transformation between reference systems.
Without a reference system, there is no meaning for the concepts of distance and time.
Any reference system for the 4-dimensional space-time must be attached to normal matter made of leptons and quarks, it cannot be attached to photons. In any reference system for the 4-dimensional space-time, the photons are particles that move with equal speeds in space and in time, while the normal matter moves faster in time than in space. The notion of proper time (i.e. the time measured for an object that moves only in time, without moving in space) is not defined for photons, because they always also move in space, not only in time.
This should be obvious from the rule introduced by Einstein that the speed of light is the same in all possible reference systems, from which the Lorentz transformations can be deduced. If a reference system were attached to a photon, in that reference system the speed of light could not have the same value as in the normal reference systems, so within Einstein's theory such a reference system cannot exist.
Consider a simpler example from basic math. Is 1/x infinite when x==0? The answer is that 1/x is undefined when x==0. In calculus one can take limits as x "approaches" 0 but x==0 is still undefined. Likewise, the Lorentz length contraction is undefined when traveling at c.
