A physics major, this is correct, in the sense that if the photons in the "matter of photons" have total energy E, then it acts like an object of mass m (under the condition that the total momentum is zero, which I presume is approximately true because the photons are moving in random directions?). However, photons do not have rest mass. The formula for photons is E = pc, where p is the momentum of the photon. The full form is E^2 = (mc^2)^2 + (pc)^2, where m is the rest mass and p is the momentum. Taking p = 0 for an object at rest gives E = mc^2, taking m = 0 for a photon gives E = pc.

Yes and no. Yes, E=mc^2 does still apply, but this refers only to a particles rest energy. So in the rest frame of a photon, the photon has no energy, and thus has no mass (or at least that's the theory, in principle more difficult to test and for all intents and purposes it is very nearly zero and most probably is exactly zero). In our rest frame, the photon moves at the speed of light, so... E = mc^2 + kintetic energy + potential energy -> which basically reduces to E = hf (plank's constant * frequency of the light).