For a photon ##E = cp##. So, he is right and you are not.

##p = mv##, as I have already explained, is a classical equation for the momentum of a particle that is not valid in relativity - for any particle, massless or otherwise.

Again, this "m" is an outdated concept of mass. It dates back to classical mechanics and is not used anymore because it is frame dependent in relativity. With the modern concept of mass (invariant mass) the equations for momentum and energy are

That equation is for a massive particle. It doesn't apply for a photon. In fact, it is not ##0## for a photon, it is undefined. Since the photon has zero mass and speed ##c## you would get:

##p = \frac{mc}{0} = \frac00##

Which is undefined. So, you have to look elsewhere for the equation that gives the momentum of a photon.

For a massless particle, you have no choice. For massive particles, the relativistic equation is always true to but the slower the particle is, the less difference there is between the results of the classical equation and the relativistic equation. At human-scale speeds, the classical equation always gives results that are useful.

In principle, the relativistic equations are always valid and the classical equations are an approximation. If that approximation is good enough, then you can apply the classical equations.

If the velocities involved are a significant proportion of the speed of light, then you definitely need the relativistic equations.

In practice, it depends how accurate you need your answer. Time calculations for GPS satellites must be so accurate that relativistic equations are needed despite the modest speeds. But, normal engineering calculations are usually more than accurate enough with classical equations.

You can always apply the relativistic equations, and you will always get the right answer. They're more exact than the non-relativistic ones.

However, the relativistic equations are also more complicated and using them is more work, so we don't use them when the relativistic effects are small enough to ignore. For example: what is the kinetic energy and the momentum of a one-kilogram mass moving at ten meters per second? Try calculating that using the classical formulas and using the relativistic ones. Which was more work? And what difference did it make?

The classical formulas can be used any time the speeds involved are small compared with the speed of light (equivalently, ##\gamma## is so close to 1 that you're OK with the approximation ##\gamma=1##). Clearly this is not the case for particles travelling at the speed of light, so you know that the classical formulas can't be used here.