NO, the oscillations in an em wave do exactly not mean that. It is a change in the field strength along some direction. The change in field strength does not mean that something is literally moving up or down in this direction. Could you please provide a reference confirming that these are indeed mechanical-like oscillations and not oscillations in the electrical field as Wikipedia and many other sources say?The point is that electromagnetic wave equation describes actual spatial wave where electric and magnetic fields oscillate, that is move "up-down"/"left-right" through actual spatial distance of their amplitudes as they propagate.
Exactly that: The electric field strength oscillates, it increases and decreases again and so on and so forth.What do you mean amplitude is "in electric field"?
Maxwell did not come up with a model for photons. He came up with a great model for light beams and large numbers of photons. Speaking about single photons (ensembles of identically prepared single photons), you can only recover some analogue to Maxwell's equations in a probabilistic manner and considering many repeated runs of an experiment. However, you still run into conceptual problems. For example, there is a weak uncertainty relation between photon number and phase. As the photon number is precisely determined for a single photon, phase is pretty much undetermined. This is something you do not get out of Maxwell's model.That's not my model, it's what Maxwell came up with. Combined electric and magnetic field and it turned out they would oscillate while propagating at the speed of light. Then Einstein figured out they have momentum, making them "full-fledged particles", to quote Wikipedia.