Is a photon emitted beyond the Earth cosmological event horizon towards Earth actually moving away from Earth? Is a photon emitted beyond the Earth cosmological event horizon towards Earth actually moving away from Earth due to space expansion? Is that the reason why we can't see beyond the horizon?
 A: The answer depends upon your frame of refence. In the reference frame of the earth, the photon is moving away, i.e. the distance from earth is increasing. In the frame of the emitter, it is moving toward the earth, but the earth is receding faster.
A: Yes, photons emitted beyond the event horizon (EH) recede from us faster than the speed of light ($c$), despite traveling in our direction. But in fact the same is true for somewhat smaller distances.
The Hubble sphere
The current distance to the EH is roughly 16.6 billion lightyears (Glyr). But with the current value of the Hubble constant, $H_0 \simeq 70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, everything farther away than $d = c/H_0\simeq 14.5\,\mathrm{Glyr}$ recedes faster than $c$. This "edge" (which isn't physical, and which isn't a "horizon") is called the Hubble sphere (HS).
However, that doesn't mean that photons emitted beyond the HS cannot reach us. As photons leave a distant galaxy, traveling in our direction but being carried away faster than they travel, the expansion of space will eventually accelerate them enough that they "overcome" this expansion, and approach us.
The event horizon
If there were no dark energy, there would be no limit to the distance from which as photon could be emitted and still reach us (this may seem paradoxical, but see the "ant on a rubber band" puzzle). But because of the the accelerated, exponential expansion there is a limit, namely the EH.
Matter at the EH is currently receding from us at $v\simeq1.15c$, so photons emitted exactly at the EH in our direction are receding from us at $v\simeq0.15c$.
The particle horizon
So, light emitted today from beyond the EH cannot reach us, but I wouldn't say that we cannot see beyond the EH, since we do see light that was emitted in the past. In fact we see all the way out to the particle horizon (PH), defined by the maximum distance that light has been able to travel sine the Big Bang. The current distance to the PH is around 46 Glyr and matter at the PH is currently receding from us at $v\simeq3.2c$.
