Is it possible that the reason photons have no mass and elementary particles do have mass is because they are created by two different types of waves, i.e. electromagnetic waves for photons and gravitational waves for elementary particles?
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No it's not as simple as this. Gravitational waves, for the purposes of this question, are the same as electromagnetic waves: they always travel at $c$ and, if / when we succeed finding a quantum theory of gravity, the particle carrying gravitation - the graviton - will be massless. We also know that it will have spin 2, as opposed to spin 1 of the photon. But that really doesn't change the spirit of this answer at all. There's even an approximation to GTR known as Gravitoelectromagnetism where gravitation follows equations that are exactly analogous to Maxwell's equations. It matches experiment beautifully (consistent with all Gravity Probe B results, for example) and the only experimental difference between it and General Relativity that I know of is that it overestimates the power lost by accelerating mass (basically because gravitational waves can only arise from quadrupole sources, not dipoles as for EM).
Be very careful with the definition of mass as "resistance to acceleration": at relativistic speeds, this resistance depends on the relative direction of the force and velocity of the accelerated object. So, in the old days, people used to think of longitudinal and transverse mass. Shove a body in the same direction as its motion, and it has resistance to inertia given by $m_0\,\gamma^3$ - rather more than its "relativistic mass", which is its total energy (rest energy together with its kinetic energy) divided by $c^2$. Shove a body at right angles to its motion, and the resistance is only $m_0\,\gamma$.
Nowadays, the only mass concept that physicists find useful is as a measure of the total energy of the body as measured from a frame that is at rest relative to the body. This is called "rest mass" or "invariant mass". Even this is awkward: invariant masses aren't additive for combined systems and are not conserved in the way total energies are. For example: two photons in opposite motion have nonzero rest mass, but the total energy of two combined systems is the sum of their total energies, less / plus any energy is released / is taken to combine them. Most of your body's rest mass and resistance to acceleration comes not from the rest mass of its fundamental particles, but from the confinement of massless gluons. When you confine a massless particle in a box, its energy adds to the system's inertia, and most mass of "matter" in the everyday world comes from this kind of mechanism.