Electron is a fundamental particle (a fermion) that carries a negative charge (1.37*10^-19C). Charge is defined as a property that can generate electric field around.
The electric field is when there is an charge present. So the question is actually very strange. If you are talking about the intrinsic properties, then you must know quantum mechanics and string theory, but it's no longer classical anymore.
And E&M wave is just a fluctuation of E&M field, so you can not ask "How the oscillation of such field creates EM wave". And photon is a quantum which is the particle property of E&M wave, again explained by quantum mechanics, photon is a wave packet.
Don't confuse the mass here with normal mass we say, because the E=mc^2 explain that if there is energy there would be an equivalent mass. So mass of photon simply indicate that photon contains energy.