But in order to have momentum, mass is required. Photons have no rest mass, but since they are never at rest, they have a measurable mass, even if that mass is purely associated with it's velocity.
No, momentum is defined as equal to sqrt(E^2-m^2*c^4)/c where m is the rest mass, E is the energy and c is the speed of light. Note that you can get a momentum out of something if m equals 0 and E does not equal 0.
But not rest mass. A photon may have finite momentum but zero rest mass. Photons aren't heavy, they're light.