hi Timmaay,
actually, E = mc^2 was a bit misleading, and might cause your confusion.
In fact, the whole equation is
E^2 = (pc)^2 + (mc^2)^2.
As you can see, it also included an extra term of momentum. Now, do not think Momentum in the classical sense, that is, p no longer equals to mv.
So indeed, it is possible for photon to not have "rest mass", it just simply means that all of its energy is contributed to momentum (or in another way, kinetic energy).
From this perspective, one could kinda see that nothing could move faster than the speed of light because some of the energy contributes to rest mass energy
Of course, this is such an interesting topic that one could go on and on and on, and deeper and deeper (wouldn't take too long for me to be out of my depth though :P). But I would like to stop at here, and turn into another way of explaining it (well, it is more of my hypothesis at this state).
Another way of thinking of it (although this I really needed to verify with my professors: that is to say, I am not 100% sure about this) was this: So for Gravitational field, Electro, magnetic field, instead of thinking of field line, one could think of mediators. i.e., for electro force/ field, the mediator is photon (and that lead to some theories of "graviton"). These photons carry some info between electrons, and photons were exchanged between electrons (to get a more comprehensive understanding, this is called QED, quantum electrodynamic. And "Particle Physics" by David Griffens had some good intro analogy of it in the, I believe the first 2 chapters).
Now, here is the place I kinda need to have verification. So if the mediators are massless, then the force/field would be inverse proportional to r^2. However, if the mediator carry even just a bit mass, then by the the law of motion, it would no longer follow the inverse square law (There are some researches conducted on this and trying to test the Gravitational field, and the possibility of the existence of graviton).
Electro field, if you recall, is proportional to the coulomb force, which was inverse proportional to r^2. (Here is the part I am not fully convinced, because there was no way that it must be.) And by QED, the mediator is photon. So to let the field be inverse proportional to r^2, photon MUST have no mass.
I hope this explains a bit if not confuses you even more. If it didn't, then simply just throw it out of the window.