I'm not sure what you mean by an "infinitesimal mass". An electron neutrino has the smallest mass of any particle I know, excepting, of course, massless particles. The electron neutrino has a mass of at most 0.3 eV.

Yes. According to theory neutrinos, photons, and gravitons all have 0 mass. (Of these neutrinos may have a very small mass... No neutrino mass has actually been measured yet. Gravitons haven't yet been observed but the theory of them predicts that they don't.) There is no problem with having a particle with no mass. Sometimes we even approximate an electron to have no mass to simplify the Math when using high energy scales where the electron mass is essentially negligible.