Celebrating James Maxwell, the Father of Light

The answer is No. The photon model doesn't come out of Maxwell's equations. You can tell, at a glance, by the absence of Planck's constant.

Digging just a bit deeper, there are some problems that can be solved using Maxwell's equations, but I don't know which of those problems already had adequate solutions before Maxwell. For instance the laws of reflection and refraction (e.g., Snell's Law) can be solved by applying Maxwell's equations at a surface boundary between two materials, but I believe they were empirically known already.

As for photons, it's actually the reverse of what you are asking for. Maxwell's equations come out of the photon model. The reason is that classical electromagnetic theory was already successful by the time that quantum mechanics emerged, so the corresponding QM theories were designed so that classical theory would be preserved as a limiting case. One of the tests of any candidate QM theory would have been: Does it preserve our existing observational evidence about the macroscopic world?

Yes: classical electromagnetism. It's an outrageously successful model of how light behaves. It doesn't predict photons, but that doesn't mean it's not an "actual" model of light.

You can include photons by taking Maxwell's equations and doing something called second quantization (https://en.wikipedia.org/wiki/Quantization_of_the_electromag...). That gets you quantum effects that can explain behaviors like shot noise, but it misses things like photon-photon scattering. (Nonetheless, I would still call this an "actual" model of light.)

Then there is full-on quantum electrodynamics (https://en.wikipedia.org/wiki/Quantum_electrodynamics), but that doesn't capture certain behaviors, like what happens to photons at the electroweak scale. But I would still call QED an "actual" model of light.