You are mixing up rest mass (which neither the photon nor the neutrino has) with moving mass / impulse
No, actually I'm not: trust me I'm a particle physicist! There are two misconceptions there. First a particle does not have to have a mass to have a momentum. Einstein's favourite equation is not actually 'E=mc^2' unless you are standing still. It is more correctly written: 'E^2=p^2c^2 + m^2c^4' where 'p' is momentum. In the case of a massless particle (m=0) this is just: 'E=pc' so a photon with a non-zero energy has a non-zero momentum but ALWAYS has a zero mass.
The second misconception is that the mass of a moving particle somehow changes. This is wrong and in fact you can show quite easily (if you know relativity) that the mass is something called a Lorentz invariant which means it is the same for all observers in all inertial reference frames. The misconception regarding the mass "getting bigger" at higher velocities comes from the formula for relativistic momentum for a massive particle "p=gamma*m*v" where 'gamma' is always greater than 1 for v>0. This factor is erroneously coupled with the mass to give what some textbooks call 'relativistic mass' (gamma*m). This is WRONG! One of the consequences of relativity is that space and time get 'mixed' differently for different observers. Velocity is 'space/time' so this is where the gamma factor comes from. This is very obvious to see if you look at acceleration. The relativistic form of Newton's second law is NOT 'F=gamma*m*a' which it would be if this was just an effect on the mass increasing.
You claim that a neutrino has always mass (or more than a photon) is either plain wrong or grants you a noble prize if you can proof it.
See this paper: “Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino Observatory”, The SNO Collaboration, Phys. Rev. Lett. 89, 1, 011301 (2002). It is not possible for neutrinos to oscillate unless they have a mass difference which means that at least two of the neutrino flavours must have a non-zero mass. You are correct that it is likely to win the authors a Nobel Prize probably within the next few years but I'm not one of them (but I was an author on the Higgs discovery paper so that's ok! ;-).